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Plate 1

FORBES CO.,B0STON.

THE

PELNCIPLE8 OF PHYSICS

ALFEED P. GAGE, Ph.D.

Author of "Elements of Physics," "Introduction to
Physical Science," etc.

Boston, U.S.A., and London

ai:N"N & COMPANY, PUBLISHERS

1897

A3
Sit

Entered at Stationers' Hall

cr ' 'N3 ^ '^

PEE FACE.

It is now thirteen years since my Elements of Physics was
published. In the interim many changes in the technical no-
menclature of Physics and many improvements in methods of
presentation of portions of the science have been made, and,
above all, the whole subject of Electricity and Magnetism has
outgrown its former apparel. Furthermore, the present is
conceded to be an era of extraordinary scientific activity, and
the demand for frequent revisions of scientific text-books is
correspondingly imperative.

The present volume, however, is a new work, and not a
mere revision of former works. Much of the material of
previous works, when suited to present needs, has naturally
been incorporated into this ; but everything has been carefully
rewritten and rearranged with reference to its adaptability to
the requirements of the present day.

The considerable increase of volume and scope of this work
over those of its predecessors may require some explanation.
In this book I have naturally been moved to attempt to meet
the demands of many highly esteemed critics who have com-
plained of serious omissions in my former works, and I am
apprehensive solely lest this volume also may fall short of the
requirements of many. Then, too, as was suggested above, the
scientific activity of this age results in successive additions
both to the theory of Physics and to its application, and thus
tends to make the ground covered by new text-books con-
tinually greater.

IV PREFACE.

But tMs book represents more than this : it represents the
author's protest against a tendency in some quarters towards
demanding ^^ smaller books," '^ cheaper books," "primers of sci-
ence," — a protest based on the conviction that true education
does not consist in the acquisition of the fewest possible facts
about any subject. Education, in Physics, implies the presen-
tation of the great truths of that science in their unmutilated
form, the indication of their relations to one another, and the
furnishing the student an o'pportunity of observing and exercis-
ing the logical processes that have led to the discovery of those
truths. Any text-book that aims to introduce the student
to a study of such importance and such inexhaustible possibili-
ties should not lose sight of this truth and encourage mere
dilettanteism. In particular there is needed a store of illustra-
tive matter, of concrete applications of general principles,
sufficient to make clear those principles and to indicate the
inductive processes by which they have been reached and the
deductions to which they lead.

Meagre information results in hazy comprehension, and
consequently provokes but meagre interest. Full and varied
treatment, on the contrary, by presenting different points of
view, clears the conceptions and thus provokes interest, and
allures to continued study. All things considered, too much
in a text-book is far preferable to too little.

In these considerations may be found a partial explanation
of the size of this volume : a work which aims to afford the
possibility and the ijicentive to more than a superficial know-
ledge of the subject.

The work contains two courses — one which is termed a high
school course, and the other an advanced course. The former Is
printed in larger type ; the latter comprises the former and
additional matter j)rinted in smaller type, which is indented
about one-fourth of an inch at the left margin of the page.
The former embraces a full course for those high schools and

PREFACE. V

academies which are able to do a fairly good work. This
course can be abbreviated at the option of teachers as neces-
sity may require.

While the advanced course does not aspire to meet the
requirements of a technical scientific course in the higher insti-
tutions, yet it is believed that, supplemented by lectures, as
all text-books should be in the higher institutions, it may meet
the requirements of the so-called classical courses in many
colleges. In the high school it provides a way for pupils
possessed of a special genius and aptitude for scientific studies
to delve deeper into many subjects than the average pupil is
wont to do.

This work is simply a text-book. It lays no claim to be a
laboratory manual. It is expected that its teachings will be
supplemented by laboratory work, for laboratory practice has
come to be considered an essential part of every scientific
course. Experiments are introduced chiefly for the purpose
of illustrating principles and laws, but tedious details which
would tend to distract the attention from, the leading facts
have been omitted.

I take this opportunity to acknowledge the debt I owe to
many able physicists and representative instructors for advice
and help. My special thanks are due to my friend, Dr. Arthur
W. Goodspeed, of the University of Pennsylvania, for very
many valuable suggestions and contributions, and for careful
reading both of the manuscript and the proof-sheets. Both
manuscript and proof-sheets have also been critically read by
Prof. Joseph 0. Thompson, of Amherst College, and my col-
league, Mr. A. P. Walker, of the English High School, Boston.
To the former I am under special obligations for much valu-
able assistance, and to the latter I am largely indebted for
whatever freedom from rhetorical and typographical errors
this book may possess.

Eor valuable criticism of proof-sheets, I have also to record

VI PREFACE.

my warm personal thanks to Dr. Daniel W. Herring, of the
University of the City of New York ; Dr. J. S. McKay, of
the Packer Collegiate Institute, Brooklyn ; Mr. C. F. Adams,
Detroit High School ; Mr. A. J. Eogers, Milwaukee High
School ; Prof. J. W. Moore, Lafayette College, Easton, Pa. ;
Mr. J. T. Coleman, Citadel Academy, Charleston, S. C. ; Mr.
A. D. G-ray, Penn Charter School, Philadelphia ; and Mr.
William Orr, Springfield (Mass.) High School. The General
Electric Company and the Zeigler Electric Company have
kindly furnished several cuts of electrical machinery and
other apparatus.

A key to the solution of problems and exercises contained
in this book will be furnished by the publishers to those in-
structors only who use this as their regular text-book.

A. P. G.

CONTENTS.

INTRODUCTION.

PAGE

Fundamental units of measurements. Kinematics. Motion, velocity,
acceleration. Laws of accelerated motion. Composition and reso-
lution of velocities. Kinds of motion 1-32

PART I.
MOLAR DYNAMICS.

Chapter I.

Force and momentum. Measurement of force. Composition and
resolution of forces. Moments of forces. Center of mass. New-
ton's Laws of Motion. Curvilinear motion. The pendulum. Work,
energy, and activity. Machines 33-117

Chapter II.
Gravitation 118-123

Chapter III.

Properties of matter. Constitution of matter. States of matter. Mo-
lecular forces. Capillarity. Diffusion of fluids 124-141

Chapter IV.

Dynamics of fluids. Transmission of pressure. Atmospheric pres-
sure. Boyle's Law. Instruments for rarefying air. Siphons and
pumps. Buoyancy of fluids. Density and Specific density . . . 142-185

Vlll CONTENTS.

Chapter V.

Energy of mass-vibration. Sound-waves. Speed of sound-waves.
Energy of sound-waves. Reflection and refraction of sound-waves.
Reenforcement and interference of sound-waves. Pitch of musical
sounds. Composition of sonorous vibrations. Vibration of strings.
Harmony and discord. Quality of sound. Analysis and synthesis
of sound-waves. Musical instruments. Vocal organs. The ear . 186-244

PART II.

MOLECULAR DYNAMICS. HEAT.

Theory of heat. Sources of heat. Temperature. Thermometry.
Calorimetry. Effects of heat : Expansion. Kinetic theory of mat-
ter. Laws of gaseous bodies. Absolute temperature. Fusion.
Vaporization. Methods of producing cold artificially. Hygrom-
etry. Diffusion of heat. Thermo-djm amies. Steam-engine . . . 245-314

PART in.

ETHER DYNAMICS.

Chapter I.

Radiant energy. Light. Speed of light. Intensity of illumination.
Apparent size of an object. Reflection of light. Refraction. Prisms
and lenses. Prismatic analysis of light. Color. Interference and
diffraction. Double refraction and polarization. Thermal effects
of radiation. Optical instruments 315-433

Chapter II.

Electrostatics. Electrification. Induction. Distribution of elec-
tricity. Electrical potential. Electrical machines. Electrostatic
lines of force. Atmospheric electricity 434-461

Chapter III.

Electrokinetics. Voltaic batteries. Some defects of batteries. Ef-
fects produced by the current. Electrical quantities and units.

CONTENTS. IX

Electrostatic units. Rules relating to an electric current. Instru-
ments for electrical measurements. Resistance of conductors.
Measurement of resistance. E. M. F. of different cells. Divided
circuits. Methods of combining voltaic cells. Verification of
Ohm's Law. Magnets and magnetism. Magnetic lines of force.
The magnetic circuit. Terrestrial magnetism. Magnetic relations
of the current. Electro-magnets. Electro-dynamics. Ampere's
theory of magnetism. Electro-magnetic induction. Dynamo-electric
machines. Electric motor. The transformer. Storage batteries.
Electrical transmission of activity. Thermo-electric currents.
Electro-magnetic theory of light. Electric radiation. Electric
light. Electrotyping and electroplating. Telegraphy. Telephony.
The bolometer. Alternating currents. Tesla's investigations . . 462-608

PRINCIPLES OF PHYSICS.

INTRODUCTION.

EUNDAMEJsTAL UNITS OF MEASUREMENT.

Accurate knowledge of physical phenomena is obtained
only by means of precise measurements of physical quan-
tities.

1. Quantity is that attribute of things which makes them
measurable, i.e. it is that which answers the questions How
much? How great ? — etc. X quantity i^ therefore a meas-
urable portion of anything.

A physical 7)ieasureme7it consists in finding how many times
a definite quantity, called a unit, is contained in the quantity
to be measured. Such a unit which has become legalized,
either by statute or by common usage, is called a standard
unit.

The expression of a physical quantity consists of a state-
ment of the concrete unit employed, e.g. pound, foot, quart,
etc., with the number of those units prefixed. The numerical
part, called the numeric, is obtained by measurement.

2. The fundamental units, in terms of which all physical
measurements are made, are those of length, mass, and time}
These are fundamental in the sense that no one is derivable

1 '■'■ The whole system of civilized life may be fitly symbolized by a foot rule, a set
of weights, and a clock." Maxwell.

2 FUNDAMENTAL UNITS OF MEASUREMENT.

from others. The three entities which these units measure
cannot be absolutely defined ; but, fortunately, they are too
familiar to require definition.

The unit of length (or ''space of one dimension") is the
meter (metric^), or yard (British). The centimeter (see p. 617)
is the one-hundredth part of a meter. A standard meter is
defined by law (1795) to be the shortest distance between the
ends of a platinum rod (made by Borda and called Metre des
Archives) at 0° C. This rod is in the keeping of the Academy
of Sciences at Paris.

The foot is one-third of the British standard yard. The
yard is defined by act of Parliament (1855) to be the dis-
tance between the intersections of the transverse lines in
two gold plugs in a bronze bar deposited at the office of the
Exchequer in London, the temperature of the bar being
62° F. (16|° C).

3. By the mass of a body is meant the quantity of matter
contained in the body. It is highly important to bear in
mind that the idea implied in the term onass is quite distinct
from that of iveight. The weight of a body changes with its
distance from the earth, while its mass remains the same.
The weight of a body is the measure of the attraction between
it and the earth, and is variable because the attraction of the
earth differs at different places, but its mass is not affected by
this attraction.

The unit of mass generally employed in science is the gram
or the pound. The gram is the one-thousandth part of the
standard kilogram. This standard is a piece of platinum,
called the Kilogramme des Archives, carefully preserved by the

1 The metric system (see page 617) is now generally employed in scientific work.
The advantage of this system consists largely in the simplicity of the relations which
exist between the standards of length and mass, and in the use of units each of
which is some decimal multiple, or sub-multiple, of the others in the same series.
"The British measurements are infinitely inconvenient and wasteful of brain-
energy." Tait. — "I look upon our English system as a wickedly brain-destroying
piece of bondage under which we suffer." Lord Kelvin.

PROCESS OF MEASURING MASS. 3

French Government at Paris. Originally it was intended to
represent the mass of a cubic decimeter of pure water at the
temperature of 4° C. A kilogram of any substance is that
quantity of the substance which, placed on a scale pan, would
just balance in a vacuum the standard kilogram placed on the
other pan.

The English unit of mass is defined by act of Parliament
(1855) to be a piece of platinum marked "P. S., 1 lb.,"
denominated the Imperial Standard Pound Avoirdupois. It
is deposited in the office of the Exchequer.

The process of measuring the mass of a body by balancing
it with a body or bodies of known mass 'is called weighing.
The process of weighing, as commonly understood, is essen-
tially a comparison of masses. A set of masses^ {e.g. a kilo-
gram down to a milligram) consists of a series of bodies
having masses corresponding to the denominations given them.

The process of measuring the mass of a body must not be
confounded with the process of finding how heavy a body is
(i.e. how great the attraction between it and the earth is),
although both processes are, in common usage, called weigh-
ing. Not only are the things measured in the two cases
entirely different, but the instruments that may be required
are quite different. For example, a kilogram mass always
has the same mass, and may, therefore, when used with the
scale balance, be relied upon always and everywhere to
measure accurately an equal mass of any body. But a kilo-
gram mass when suspended frorn a spring balance may be
found to be heavier in certain situations than in others, hence
this instrument does not under all circumstances measure
mass correctly. On the other hand, the scale balance will
not detect any change in the heaviness of a kilogram mass ;
hence this instrument does not always measure heaviness
correctly. For most practical purposes, however, these in-

1 Commonly called a set of weights.

4 FUNDAMENTAL UNITS OF MEASUREMENT.

struments may be used intercliangeably, inasmuch as at the
same place mass is proportional to weight.^

The unit of time generally employed in scientific meas-
urement is the second. The second is „ „1^^ of the mean

o b -l U U

solar day.

4. Derived units ; C. G. S. System. — The system of measure-
ments in which all the units used in measuring physical quantities
are derived from the three metric units given above is called the
centimeter-gram-second system, or, briefly, the C. G. S. system.
The system is also called an absolute system of units.

In any absolute system the unit of length is represented symbol-
ically by [L], that of mass by [M], and that of time by [T]. Any
derived unit may be represented by certain pov^ers of these symbols,
or by the product of certain powers of these symbols. Thus the
unit of area = [L^].

5. Dimensional equations. — Any equation showing what powers
of the fundamental units enter into the expression for the derived
unit, is called its dimensional equation. The dimensional equation
for any derived unit is deduced from the physical laws by which
the unit is defined. /

6. Volume. — By the volume of a body is meant the quan-
tity of space it occupies. The derived unit of measurement
of volume is the cubic centimeter (or cubic foot), and is
defined as the volume of a cube the length of one side of
which is one centimeter.

The dimensional equation for volume is (V) = [L^].

7. Matter, body, substance. — Provisionally we may define
matter'^ as that which occupies space. A body is any limited

1 This is one of many instances in physics in which one quantity is indirectly
measured by measuring another proportional to it. Legitimately speaking it is not
the function of a spring balance to measure mass, nor of the scale balance to
measure heaviness. The first measures stress (§ 63) ; and the second demonstrates
equality of moments (§ 50).

2 Matter is variously defined in scientific text-books according to the fancy of the
authors. The definitions are, however, only provisional, serving merely the practical
requirement of distinguishing between what is matter and what is not matter. The
question "What is matter?" is still a subject of pure speculation, and its discussion
therefore wholly unsuited to a scientific text-book. The discovery of its ultimate

PHYSICAL PHENOMENA AND PHYSICAL LAWS. 5

portion of matter, e.g. a lake, a tumbler, a desk, etc. Names
of bodies should be carefully distinguished from names of
substances which indicate merely the kind of matter of which
the bodies are composed, such as water, glass, wood, etc.

8. Phenomena, physical laws. — By observation we learn
that a piece of iron expands when heated, begins to be
luminous when heated to a certain temperature, and changes
to a liquid when heated much more. Here are three distinct
changes which heat may effect. Changes or events like these,
and countless others which occur in nature, are called physical
phenomena.

If, after many trials with iron from various sources and
under varying conditions, we are able to state in general that
the application of heat in suitable quantities to iron will be
followed by these phenomena, such a generalized statement is
called a physical law. A physical law is an expression of a
constant relation ivhich has been discovered to exist between
certain physical quantities.

Physical laws, unlike statute and moral laws, do not govern
events, but are generalized statements of the order of events.
Iron does not expand in obedience to a law that " heat expands."
We do not explain why a body falls to the earth by stating a
law that " an unsupported body falls"; indeed, the cause of a
fall has never been discovered, though every one of us has
individually discovered the law just quoted.

nature may be beyond the range of human intelligence. The following attempt to
answer this question is given (1) to illustrate the significance of the question ; (2) as
a statement of the theory probably entertained at the present time by our most
advanced scientists : — " Matter is the rotating parts of an inert perfect fluid which
fills all space, but which is, when not rotating, absolutely unperceived by our senses."
Lord Kelvin.

KINEMATICS.

Sectiojst I.

MOTION, VELOCITY, ACCELERATION".

9. Motion. — Kinetnatics (from Ktvyjixa, motiofi) treats of
motions without reference to their causes. Motion is a con-
tinuous change of position. The position of a particle of
matter is determined by its direction and distance from another
particle, or from some point of reference. A particle moves
relatively to a given point while the straight line con-
necting it with the point changes either in directio7i or
length. A particle is at rest relative to a given point while
a straight line joining them changes neither in direction nor
length.

While you are opening or shutting the legs of a pair of
dividers (A, Fig. 1), a straight line a^ V connecting the points
at the ends of the legs changes in length ; hence there is
relative motion between these points. If (B, Fig. 1) you
open the legs a little way, and, fixing the end of one of the
legs upon a plane surface, trace a circle with the end of the
other leg around the former as a center, there will be relative
motion between the two points, since a line joining them,
ah, ab', etc., changes m direction.

If (C, Fig. 1) you trace with the points of the open dividers
two straight parallel lines on a plane surface, the two points
will be relatively at rest, just as surely as if the dividers were
lying upon the table, since in both cases a straight line con-

RELATIVE MOTION. 7

necting the points a b, a' b', etc., changes neither in length nor
in direction.

A point may be at the same instant at rest with reference
to certain points, and in motion with reference to certain other
points. For example, while the points of the dividers are
tracing straight lines on the plane surface (C, Fig. 1), and are
relatively at rest, they are in motion with reference to every
point in the plane surface.

When a particle is spoken of as being in motion or at rest, some
point is always expressed or understood, relatively to which the
change or permanence of position is maintained. All motions with
which we have to do, or which we can measure, are relative. Change
of position with reference to a fixed point in space would be absolute
motion. But there is no fixed point which we know. " There are
no landmarks in space ; one portion of space is exactly like every
other portion, so that we cannot tell where we are" or in what
direction, or how fast, we are going.

In ordinary language the phrase " a body at rest " means that the
body does not change its position with reference to that on which it
stands, as, for instance, the surface of the earth or the deck of a

8 KINEMATICS.

ship. It can mean nothing else, for both it and all points of the
earth's surface are in rapid motion with reference to the sun and
other heavenly bodies, and also with reference to the earth's axis.

A body moves as a whole with reference to any point only
when a certain point of that body, called its center of mass,
or centroid,^ changes its position with reference to the given
point. Thus the relative motion of two bodies is determined
by the change of position of their centroids. Likewise the
path in which a body moves should be understood to mean the
line described by its centroid.

10. Velocity/. — No motion is instantaneous. A body con-
sumes time, longer or shorter, in its transit from one position
to another. Eate of change of place (or time-rate of displace-
ment) of a body is called its velocity. Velocity is expressed by
stating the number of units of distance traversed in a unit of
time. For scientific purposes it is most frequently expressed
in centimeters per second ; for practical purposes the units of
time and distance are chosen at convenience, as the velocity
of a locomotive in miles ijer hour, of a rifle bullet in feet per
second, etc. Observe that velocity is dista7ice per unit of time,
and cannot be correctly expressed in miles, feet, etc., alone.

Velocity involves the idea of direction, and may change in
both magnitude and direction. It is sometimes convenient to
ignore the direction of a body's rate of displacement, in which
case we use the term speed. Thus it is better to speak of the
speed of the locomotive and the bullet when only the magnitude
of the change in position is considered.

11. Constant and accelerated velocity. — When a body moves
in a straight line with unchanging rate, i.e. when it traverses
equal spaces in equal times, its velocity is said to be constant.
In case, however, of a continuous increase or diminution of

1 Both the expressions "center of mass" and "center of gravity" are open to
objections ; hence certain careful writers have suggested as a substitute the term
centroid.

CONSTANT AND ACCELERATED VELOCITY. 9

velocity, it is said to be accelerated. Finally, if this growth
or diminution of velocity is uniform, it is said to have constant
acceleration.

When the velocity increases, as in the case of a falling
stone, its acceleration ^ is said to be positive, or -f- ; when the
velocity decreases, as in the case of a stone thrown upward,
its acceleration is said to be negative, or — .

Velocity is determined by dividing the distance traversed

by the time consumed. If a body move s feet in t seconds, its

s s

velocity, v, is - feet per second, or ?;==-. In case the velocity

be accelerated, this result is to be regarded as the average
velocity for that distance ; and in the case of uniform motion
the average velocity is the same as the actual velocity at every
instant. It is evident that the actual velocity of a body whose
motion changes can be given only at some definite instant or
point in its journey. It denotes the space which tvould be
traversed in a unit of time, if at the given instant the velocity
should become constant.

In the C. G. S. system the unit of velocity is that rate of dis-
placement at which a unit of length is traversed in a unit of time,
and the unit chosen is a centimeter per second. Its dimensional is
[L/T, or LT-i].

The change of velocity of a particle per unit of time is called
its rate of acceleration, or simply its acceleration, and is
represented by a. When a particle acquires equal changes
of velocity in equal units of time its acceleration is said to be
constant, and its motion uniformly accelerated. The accelera-
tion of a body falling in a vacuum, and of a body projected
vertically up in a vacuum is practically constant ; in the
former case it is about 32.2 feet (or 9.8 m) per second, in the

1 Acceleration etymologically means an increase of speed, but for convenience
it has lately come to be applied in scientific treatises to either an increase or a
decrease of speed.

KINEMATICS.

latter case it is a negative acceleration of about 32.2 feet
per second.

The average acceleration, a, of a particle in traversing a
certain distance in a given time, and tlie rate of acceleration
(also rej)resented by tlie symbol a), provided the acceleration
is constant, is found by dividing the entire change in velocity,
V, in a certain time by the time, t^ taken in making the change,

i.e. a = -, whence v = a t. Thus, if the velocity of a rail-
road train at a certain instant be 25 miles per hour, and
half an hour hence it be 15 miles per hour, then the entire
change of velocity, v, is — 10 miles per hour ; hence the
average acceleration, i.e. the acceleration if it were uniformly

distributed throuarhout the 30 minutes, is ^,, =( ] oi a

30 ^ 3^

mile per minute. Again, if a stone falling with a constantly

accelerated velocity acquire in 4 seconds a velocity of 128.8 feet

128 8
per second, its acceleration is '^ ' = 32.2 feet per second.

In the C. G. S. system the unit of acceleration is that acceleration
in which a unit of speed is gained or lost per unit of time. Its
dimensional is [LT~2].

Section II.

LAWS OF UNIFORMLY ACCELERATED MOTION.

12. First Lata. — If a particle be moving . at a certain
instant at a rate V, and its acceleration be + <^? then its
velocity, v, at any instant is expressed as follows :

At the initial instant, v = F ;

At the end of the first unit of time, v=V-\- a X 1 ;

At the end of two units of time, v ^V-\- a X 2 -,

At the end of t units of time, v ^V-\- at (A).

The last, (A), is a general equation expressing the relation

LAWS OF UNIFOKMLY ACCELERATED MOTION. 11

between the velocity (v) at the end of any given unit of
time (t), the original velocity (V), and the acceleration (a).
Erom this formula we derive the following law :

(1) Change of velocity due to uniform acceleration is equal to
the product of the acceleration and the units of time. Hence the
change of velocity is proportional to the rate of acceleration,
and to the time occupied.

13. Second Laiu, — If its initial velocity, V, be zero, i.e. if
the particle start from a state of rest, the equation becomes
v = at.

Since the velocity of a particle starting from a state of
rest increases from zero to a t, the average velocity must be

— - — =^^at. At this rate in the same time, t, it would

traverse a distance, S, equal to ^ at X t^=^ at''^ units ; hence
8=^^ a t^ (B). Prom this we derive the law: (2) The distance
traversed in a given time by a particle starting from a state of
rest and having uniformly accelerated velocity., is one half the
product of the acceleration and the square of the units of time.
Hence the entire distance traversed is proportional to the
square of the time, and to the acceleration.

If a particle, instead of starting from a state of rest, have
an initial velocity, F, it would move in t units of time without
acceleration a distance V Xt \ to this distance must be added
the distance it moves in consequence of acceleration, in order
to obtain the entire distance traversed in t units, and our
formula becomes S^^Vt^^at'^ ( 6') .

14. Verification. — The two laws given above are verified
approximately and conveniently by the use of the venerable
Atwood's machine.^ The equal weights A and B (Fig. 2) are
suspended by a thread passing over the 'wheel C. Inasmuch
as the weights are equal they counterbalance each other and

1 This macMne is a contrivance which enables lis to increase the mass to be moved
without increasing the force which moves it, thus so decreasing the acceleration as
to render approximate measurements feasible.

KINEMATICS.

remain at rest. Raise the weight A and
place it on the platform D as shown in Fig. 3.
Place on this weight a small additional one,
E, called a " rider/' the weight of which sets
the system in motion. Set the pendulum F
swinging. At each swing it causes a stroke
of the hammer on the bell G. At the instant
of the first stroke the pendulum causes the
platform D to drop so as to allow the weights
to move. When the weights reach the ring H,
the rider, not being able to pass through, is
H caught off by the ring. Eaise and lower the
ring on the graduated pillar I, and ascertain
by repeated trials the average distance the
weights move between the first two strokes
of the bell, i.e. during one swing of the
pendulum. Inasmuch as all swings of the
pendulum are made in equal intervals of time,
we may take the time of one swing as a unit
of time. We will also, for convenience, take
for a itnit of distance the distance the weights
move during the first unit of time, call this
unit a space, and represent the unit graphically
by the line a h (Fig. 4).

Next ascertain how far the weights move
from the starting point during two units of
time, i.e. in the interval of time between
the first and third strokes of the bell. The
distance will be found to be four spaces,
or four times the distance that they moved
during the first unit of time. This distance
is represented by the line a c.

Now ascertain the velocity which the
weights have at the end of the first unit of

Fig. 2.

LAWS OF UNIFORMLY ACCELERATED MOTION.

time. Place the ring H at the
point (h) which the weights have
been found by trial to reach at
the end of the first unit of time.
Allow the weights to descend as
before. At the end of the first
unit of time the rider is caught off.
At this instant acceleration ceases
and the motion becomes uniform.
Ascertain how far the weights move
with uniform velocity during the
second unit of time ; this velocity
is evidently the velocity which the
weights have at the end of the first
unit of time. This distance will be
found to be (approximately ^) two
a

Fig. 3.

1 U.of T. h.

2 U.of T.

3 U.of T.

1 space.

Represents the velocity at the end of the'
of time ; also the acceleration during
unit of time.

first unit
the first

Velocity at the end of the second unit of time.
Acceleration during the second unit of time.

Velocity at the end of the third unit of time.
Acceleration during the third unit of time.

4 U.ofT. e.

Fig. 4.

1 Approximately, since they are retarded by the resistance of the air and the
friction of the wheel.

14 KINEMATICS.

spaces ; hence tlie velocity at the end of the first unit of time
is two spaces per unit of time. But the velocity at the beginning
of the first unit of time was zero, hence the acceleration during
the first unit of time is ttvo spaces per unit of time.

In like manner determine the velocity at the end of the
second unit of time. It will be found to be four spaces per
unit of time. And as the velocity at the end of the first unit
of time was two spaces per unit of time, the acceleration
during the second unit of time is two spaces per unit of time.
Hence the acceleration during the first two units of time is
uniform, and the change of velocity during the first two units
of time, as stated in law (l), = a^ = 2 X2 = 4 spaces per unit
of time.

Exercises.

1. The velocity of a particle at a certain instant is V ; its acceleration
is a ; what will be its velocity, -u, in t units of time afterward ?

2. If the initial velocity of a body be F, its acceleration a, and its
final velocity v, how long, ^, was it in acquiring its final velocity ?

3. If a body having an initial velocity V acquire in t seconds a velocity
V, what was its acceleration ?

4. If a body move from a state of rest with a uniform acceleration a,
what space, /S, will it traverse in t units of time ?

5. If a body move from a state of rest with an acceleration a, in what
time, t^ will it traverse the space S ?

6. The velocity of a particle at a certain instant is 20 feet per second ;
its acceleration is 3 feet per second ; what will be its velocity 10 seconds
hence ?

7. Suppose that the acceleration of the particle mentioned above is
— 2 feet per second, what will be its velocity 5 seconds after the instant
named ?

8. a. A body falls from rest ; its velocity increases (if we disregard
the resistance of the air) 32.2 feet per second. What is its velocity at
the end of the first second ? 6. What, at the end of the tenth second ?
c. What, at the end of half a second ?

9. If the initial velocity of a body be 5 feet per second, its final
velocity 26 feet per second, and its acceleration 2 feet per second, what
was the time consumed in acquiring the final velocity ?

EXERCISES. 15

10. A bullet is projected vertically upward with an initial velocity of
161 feet per second ; what will be its velocity at the end of the third
second (a = — 32.2 feet per second) ?

11. How long will the bullet named in the last problem rise ?

12. What velocity will the bullet have at the end of the sixth second,
and in what direction will it be moving ?

13. a. What distance will a body fall from a state of rest in one
second ? b. In two seconds ? c. In ten seconds ?

14. A stone thrown vertically downward is given an initial velocity of
40 feet per second. How far will it descend in ten seconds ?

15. a. A bullet is projected vertically ujDward with an initial velocity
of 225.4 feet per second ; how long will it rise ? b. How far will it rise ?

16. How long will it take a body to fall 1030.4 feet from a state of
rest?

17. a. A body falls during 1^ seconds ; what is its final velocity ?
b. How far does it fall ?

18. A body falls 297.6 feet in 4 seconds ; what was its initial velocity ?

19. What initial velocity must be given a body that it may rise 6
seconds ?

16 KINEMATICS.

Section III.

COMPOSITION AND RESOLUTION OF VELOCITIES.

15. Gra]}hical representation of ^motion and of velocity. —
If a person wish to describe to you the motion of a ball struck
by a bat, he must tell you three things : (1) ivhere it starts,
(2) in what direction it moves, and (3) how far it goes. These
three essential elements may be represented graphically by a

straight line. Thus, suppose balls

at A and D (Fig. 5) to be struck by

D ^ E bats, and to move respectively to

B and E in one second. Then the
points A and D are their starting-points ; the lines A B and
D E represent the direction of their motions, and the lengths
of the lines represent the distances traversed. In reading, the
direction should be indicated by the order of the letters, as
AB and DE. The lengths of these lines are not equal to the
distances traversed by the two balls, but represent these
distances drawn to some convenient arbitrary scale ; thus on
a scale of 1 cm = 10 m, these lines represent distances of 32
and 20 meters respectively.

The velocity of a moving body is described by giving (1) its
directioii, and (2) the units of distance per unit of time. Since
the lines AB and DE represent the distances traversed by
the two balls during the same unit of time, these lines like-
wise represent their average velocities during this time, i.e. A B
represents an average velocity of 32 m^ per second, and D E an
average velocity of 20 m per second.

16. Composition of simultaneous velocities. -, — If a particle
have by any means two or more separate and independent
motions communicated to it simultaneously, and if the motions
imparted be themselves constant in velocity and direction, the
result of their concurrence is a single motion in a straight line

COMPOSITION AND RESOLUTION OF VELOCITIES. 17

with a single velocity and direction. This is illustrated some-
what imperfectly in the following manner. With the handle A
in the position shown in Fig. 6, push it forward carrying the
frame B C to the right. This frame carries a pencil D whose
point presses the paper below, and as the frame advances, the
line a b is traced upon the paper, graphically representing the
motion of the pencil. If, when the pencil point is at a and
the frame is at rest, the string G be pulled, the pencil will
trace the line ac at right angles to ab. Kow these two
independent motions may be communicated to the pencil
simultaneoiisly by fastening the string E to the binding screw F
and pushing forward the handle A. The pencil point will not

move in either of the lines ab or ac, but its motion will be
intermediate between the two, and it will trace the line a d.
This single motion, which is the result of the concurrence of
two motions, is called their resultant ; and they, with regard
to the resultant, are called its comioonents.

The distance a cl is traversed in exactly the same time that
the distance ab would be traversed if the pencil had no
other motion, the handle A being pushed forward with the
same speed in both cases ; likewise the distance ad is
traversed in the same time that the distance (xc is accomplished,
when the string is simply pulled over the pulley G with the
same speed, and has no other motion. The lines ab, ac,
and ad represent not only the distances traversed in the

KINEMATICS.

several directions, but also the magnitudes and directions of
the respective velocities. For example, if the velocity be

constant and the pencil reach
successively at the end of equal
intervals of time the points 7?^",
n^', and d (Fig. %a), then am",
?/2," 7i'\ and n" d represent its
velocities in the successive inter-
vals, and a m, m n, and n h repre-
sent the velocities for the same
intervals in the direction a h ; and a m', m' n\ and n^ c the
velocities in the direction ac.

If points G and d, and d and h are joined by (dotted) lines,
we have a parallelogram of which the line ad, representing
the resultant, is a diagonal. Hence to find the resultant of
two simultaneous velocities when they make an angle with
each other, the rule is : Construct a parallelogrcun of ivhich
the adjacent sides represent the two velocities, and the
diagonal ivhich lies between these adjacent sides rejjresents their
resultant.

When more than two com-
ponents are given, find the re-
sultant of any two of them, then
of this resultant and a third,
and so on until every component
has been used. For example, let
the several velocities imparted ^,
to a particle be represented by
the lines AB, AC, AD, and
^^ (Fig. 7). The resultant of
A B and ^ C is ^ F; the result-
ant oi AF and AD i^ A G ;
that oi AG ?iTidi AF i^ AH
which represents the resultant of the four velocities.

RESOLUTION OF A VELOCITY INTO COMPONENTS. 19

When two components are at right angles to each other, it
is evident that we may obtain the magnitude of the resultant
by finding the square root of the sum of the squares of the
two components.

In case a particle has several velocities imparted to it, all
in the same direction, their resultant is the sum of all. If
some are opposite others, one of the two directions is con-
sidered as positive and the opposite direction as negative,
and these signs being prefixed to the numerical values, their
algebraic sum is the resultant.

17. Resolutio7i of a velocity into components. — Any motion
or velocity may be resolved into two or any given number of
motions of velocities. Let A B (Fig. 8) repre-
sent the velocity and direction of motion of a
particle. Draw a line AC to represent, either
arbitrarily or according to the conditions of
the problem, one of the required components.
Connect B and (7, draw A D parallel with B C,
and D B with A C, and thus complete a '^^^' ^'

parallelogram of which ^^ is a diagonal. The two adjacent
sides A C and AD represent two component velocities of
the particle ; in other words, a particle having a velocity
represented by the line AB has at the same time velocities
represented in magnitude and direction by the lines A C
and AD.

Exercises.

1. a. If a ship move east at the rate of 10 miles an hour, and a person
on deck walk towards the bow at the^rate of 2 miles an hour, what is the
resultant of these two velocities ? h. With reference to what has he this
velocity ?

2. Suppose the person mentioned above walk aft at the rate of 2 miles
an hour, what will be the resultant of these two velocities ? 6. Prefix
suitable signs to the numbers given and represent the addition v^^hich
gives the resultant.

20 KINEMATICS.

3. a. A particle moYes simultaneously northward with velocity a, and
southward with velocity 6 ; what is the resultant of these velocities ?
6. How do you interpret the resultant if a >► 6 ? c. How, if 6 >► a ?
d. How, if a =6?

4. Suppose the person mentioned above walk directly north across the
deck at the rate of 4 miles an hour, what will be the resultant of these
two velocities ?

5. Suppose the person walk northeast at the rate of 4 miles an hour,
what will be his resultant velocity ? [In drawing the parallelogram of
velocities, represent the component velocities to some scale, e.g. I of 1
inch or 1 cm = 1 mile, then having completed the parallelogram and
having drawn the diagonal which represents the resultant, measure the
latter and the result will express, on the scale chosen, the resultant
velocity required.]

6. Suppose an attempt be made to row a boat at the rate of 6 miles an
hour directly across a stream flowing at the rate of 10 miles an hour ;
determine the direction and velocity of the boat.

7. A vessel sails south-southeast {i.e. 22.5° east of south) at the
rate of 14 miles an hour ; determine its southerly and its easterly
velocity.

8. Represent graphically, to scale, a velocity of 100 feet per second
and resolve this velocity into two components which shall have an angle
between them of 45°.

9. Represent graphically velocities, all in different directions, which a
particle has at a given instant, as follows : 20 feet, 30 feet, 15 feet, and
25 feet, per second. Determine its apparent velocity and direction.

.18. Composition of constant ivith accelerated velocity. —
Experience teaches that a body, e.g. a stone, projected in a
horizontal direction moves not in a horizontal path, but in a
path intermediate between a horizontal and a vertical one,
showing that its velocity is composed of a horizontal and a
vertical component. Its horizontal velocity (if the resistance
of the air be disregarded) is censtant and its vertical velocity
is uniformly accelerated. Let AB (Fig. 9) represent the verti-
cal component of the motion during the first second, then B C
and C D will represent its vertical motion during the second
and third seconds respectively. Let AB', B'C, and CD' rep-
resent successive horizontal motions during the same three

CONSTANT WITH ACCELERATED VELOCITY.

periods. Then it is evident by a combination of these two
motions that the body will pass from A to B" during the first
second, from B" to C" during the second second, and from C"
to D" during the third second. The body traverses a curvi-
linear path called a parahola, as shown in the figure. In
practice, the resistance of the air would modify the nature

of the curve somewhat, so that its real path is a peculiar
curve known in the science of gunnery as a hallistic curve or
trajectory.

It should be borne in mind that one of the component
velocities of a particle moving in a curvilinear path is always
accelerated.

Problem. — Imagine a body to be projected obliquely upward at an
angle of 45° ; represent arbitra.rily its vertically downward accelerated
motion, and its obliquely upward constant motion for three seconds,
and determine the actual path traversed by the body during this
time.

22 KINEMATICS,

Section IV.

KINDS OF MOTION.

19. Motion of translation and rotation. — In pure motion of
translation all the points of a body move with the same
velocity and in the same direction (Fig. 10). Example : the

Z X_-A

Fig. 10. Fig. 11.

Rectilinear motion of translation. Motion of rotation.

motion of an elevator or a piston in the cylinder of a station-
ary steam engine. When the points of a body describe arcs
of circles having its centroid for a common center, the motion
is one of pure rotation (Fig. 11). Example : the motion of a
wheel or a top. All possible varieties of motion may be pro-
duced by the combination of translation and rotation (Figs. 12
and 13). Examples : the motions of the planets, that of a
ball thrown from the hand, that of a carriage wheel along
a road.

When a body rotates, every particle in the body describes a
circle around some point or line which is the center or axis of
rotation.

The velocity of a point far from the axis is greater than
that of a point nearer the axis, and, generally, the velocity
of a point is proportional to its distance from the axis ;
hence the expression "velocity of a rotating body" is mean^
ingles s.

ANGULAR VELOCITY.

20. Angular velocity. — We may, however, speak of the angular
velocity of the rotating body, which is the same for all points in the
body. Angular velocity is rate of rotation, and is measured by the
angle turned through by the rotating body in any given unit of time.

The unit angle in terms of which angular velocity is measured is
called a radian, and is the angle which is subtended by an arc equal
in length to the radius.

It is customary then to express angular velocity of a body in
radians per second, and this is numerically equal to the speed of a

Fig. 12.
Translatory and curvilinear motion.

Fig. 13.

Combination of translatory and

rotary motion.

point one unit distant from the axis. The Greek letter w (pro-
nounced o-meg'-a) is chosen to represent angular velocity.

Now since the linear velocity of any point in a rotating body is
proportional to its distance, r, from the axis, it must be the product
of the angular velocity of the body multiplied by the distance of the
point from the axis, i.e. v = r u.

It remains to develop a formula for finding the value Of w in terms
of the time of a complete rotation. It is known that the ratio
between the circumference of a circle and its diameter is about 3^,
and is usually represented by the Greek letter tt. Now if the

24 KINEMATICS.

body make one rotation in T seconds, the angular velocity is
equal to the total angle, 2 tt radians, divided by the time, T,

or co = — radians per second.

Questions.

1 . A body makes ten rotations per second ; what is its angular
velocity ?

2. What is the actual velocity of a point in this body ten inches
from the axis of rotation ?

3. a. What is the angular velocity of the earth's rotation ?
6. Does its angular velocity vary at different latitudes ? c. Does
the actual velocity of points on the earth's surface vary at different
latitudes ?

4. What kind of motion is that of the earth in its orbit ?

5. Why is it meaningless to speak of the velocity of rotation of a
body?

6. a. What motions have the wheels of a carriage drawn straight
along a level plane ? h. What motion has the carriage ?

7. Compare the several velocities of the small front wheels of the
carriage with those of the larger hind wheels.

8. If a wheel make 200 i^evolutions per minute, what is its
angular velocity ?

21. Rectilinear and curvilinear motion. — Besides change in
velocity or rate of motion, there may be a change in direction
of motion (see p. 6). When a particle moves in a constant
direction, i.e. in a straight line, as in the case of a freely fall-
ing bnllet, its motion is said to be rectilinear. Bnt if its
motion constantly changes in direction, i.e. at every point, aS'
is the case of every particle in a rotating wheel except points
on its axis, its motion is said to be curvilinear. It is
evident that the direction of a motion in a curvilinear path
can be given only for some specified point, and, further-
more, that direction can be represented only by a straight
line, for a curved line is a line composed of an infinite
number of directions. Let A (Fig. 14) represent a body
mounted on a cardboard sector S S^ which is rotated about

co:mposition of circular motion.

tlie axis C in the direction indicated by the arrow. The
body will move in the circular path ADEF. The straight
line AB will indicate the direction of the motion at every
point, but it will be seen that this line changes its direction
constantly. At whatever point the body may be at any
instant, the line A B, which shows the direction of the motion,
is always tangent to the curve at that point.

Fig. 14.

Fig. 15.

22. Composition of circular motion. — If a particle move in
a circular path, e.g. a stone whirled in a sling, its motion
every instant is the resultant of a tangential motion and a
centripetal (toward the center) motion. If when it passes
point A (Fig. 15) its tangential velocity be represented by A B,
its centripetal velocity may be represented by B C, because at
the end of the unit of time in which it would reach B if it
were moving in a straight line, it is found to be not at B, but
at some other point, C, nearer the center by the distance B C.

23. Simple Harmonic Motion. — Besides the motions of
translation and rotation already considered, a third kind of
motion must be studied somewhat in detail, as it plays so
important a part in subjects to follow, notably sound and
radiation.

KINEMATICS.

Fig. 16.

If a lead bullet A (Fig. 16) be sus-
pended by a thread and set swinging
in a horizontal circular path, the
motion is practically uniform, and,
when viewed directly from above or
below, appears circular. But if the
motion of the bullet be viewed by
the eye placed on the same level
with it, it seems to travel to and fro
in a straight line, but with varying
speed. It is seen to move slowly
near the ends of its path and more
rapidly in the middle. The motion
of the bullet as now viewed is virtu-
ally the projection of uniform cir-
cular motion on a diameter.

Thus, let Fig. 17 represent a particle moving with uniform
speed in the circle ADMN, as in the case of the swinging
bullet. A line drawn from this particle (at the instant it
passes the point A in the
circle) perpendicular to the
diameter M N, intersects it
at the point a. While the
particle moves to B, C, D,
E, F, and M, this intersect-
ing point moves to ^, c, d^
e,f, and M. Although the
speed of the moving par-
ticle is uniform, moving
over the equal spaces A B,
B C, CD, etc., in equal
intervals of time, the speed
of the intersecting point is
variable, moving in the same intervals of time respectively

Fig. 17.

SIMPLE HARMONIC MOTION. 27

through the unequal spaces ab, be, cd, etc. The speed of this
point is greatest as it passes the center of the circle,
diminishes toward the extremities of the diameter M N, and
is momentarily zero at the points M and N. Such a motion
as that, forward and backward along the line MIST, executed
in equal periods of time, is called simple harmonic motion, or
S. H. M.

It is the kind of motion executed by the vibrating prongs
of a tuning fork, by a stretched string when emitting a
musical sound, by particles of air when traversed by sound
waves, and very nearly by a pendulum swinging in a short
arc. A like motion occurs in the ether when traversed by light
waves or electrical waves, and in every elastic medium when
set in a tremor ; hence its intimate relation with many
branches of physics.

As the motion repeats itself in regular intervals, it is said
to be periodic. The time occupied by a particle in executing
a single complete harmonic motion, i.e. from M to IST and
back, is called a period. The period is, evidently, the time
occupied by one complete revolution in the circle of reference,
as N, D, M, C , ]Sr. When the body appears to move from left
to right its motion is said to \)Q positive ; and when from right
to left, negative. The extent of the vibration on either side
of its middle point, as M or IST, is called the amplitude of
the S. H. M. The distance of a moving particle from the middle
point at any instant, as e, is called its displacement, and
points M and N are points of greatest elongation. The position
of the particle at any instant is denoted by its phase, which
is defined as the fraction of a period since the particle last
passed through in the positive direction.

24. Composition of Simple Harmonic Motions. — Simple harmonic

motions in the same or different directions may be compounded

according to tlie same laws as uniform motions, accelerations, etc.

We consider first the composition of S. H. M.'s of the same period.

The resultant of two or more S. H. M.'s of the same period and in

KINEMATICS.

the same direction is a S. H. M. of the same period as that of the
components and having an amplitude equal to the sum of the
amplitudes of the several component motions. This assumes that
the phases are the same. If, hov^ever, a particle be subjected to
two S. H. M.'s along the same line, but differing in phase by half a
period, the resultant will be a motion with an amplitude equal
to the difference of the amplitudes of the components. That is,
the particle will remain at rest if the component amplitudes
be equal.

If the two motions be in the same phase or in opposite phases
(i.e. having a difference of phase of one-half of a period) and at
right angles to each other, the amplitude of the resultant will be
represented by the diagonal of a rectangle constructed upon those

lines as adjacent sides which rep-
resent the amplitudes of the com-
ponent motions. Thus in Fig. 18,
let A B and C D represent two com-
ponent S. H. M. 's at right angles to
each other, the center of the circle of
reference being at 0. Then E G or
r H will represent the resultant mo-
tion if the phases are alike or oppo-
site.

If the difference of phase be only
i period, then the resultant motion
will be a circle, or an ellipse, accord-
ing as the component amplitudes are equal or not. If the difference
of phase be a fraction of a period not as simple as J or i, then the
resultant motion in general will be an ellipse with its center at 0,
the position of its major axis depending on the relation of the com-
ponent amplitudes and on the difference of phase.

Experiment 1. — The principles given above may be verified
experimentally by the use of an approximately simple pendulum.
Partly iill a small glass funnel (Fig. 19) with fine dark writing sand
and suspend it from a frame above by means of a string (say) 1 m
long. It can be made to oscillate at will in any direction and the
sand falling upon a horizontal surface, e.g. a large sheet of white
paper or pasteboard, leaves a tracing of the motions of the pendulum.
The position of rest of the bob being at (Fig. 18), give it an impulse
with the hand in the direction OA sufficient (say) to carry it to

COMPOSITION OF S. H. m's. 29

A. We may, for convenience, divide the complete motion into four
parts, each of i period, e.g. from to A, A to 0, to B, and B to
0. A similar impulse may be given to the bob in any other direc-
tion, for example in the direction C. The motions of
such a pendulum are approximately S. H. M.'s. The
directions A and C will be considered as positive,
and their opposites, B and D, negative.

We will now examine four cases : (1) While the bob is
oscillating in the path A B, gently tap it with the hand
in the direction C at the instant it passes in the positive
direction ; the bob will now move along the line E, the
diagonal of the square on A and C. In this case the
component motions are in the same phase, since both
impulses are given at and in positive directions.

(2) While the bob is moving through toward A, let
the second impulse be in the direction D ; the resultant
motion is now in the direction H, and the bob oscillates '■
along the line F H without change of period. In this ^^^- ^^^
case the phases are opposite, differing by i period.

(3) Let the impulse be in the direction C, but applied when the
bob is at A instead of at 0. The motion now is in a circle, the
direction being anti-clockwise (i.e. opposite to the direction in
which the hands of a clock move). Here the component B A is ^
period ahead of the component D C.

(4) If the impulse be given at A in the direction D, the resultant
will be a circle still, but traversed in the opposite direction, the
component B A being now J period behind the component D C.

Observe that circular motion may be regarded as compounded of
two S. H. M.'s at right angles to each other, the phases differing by
J period.

If the component amplitudes be not equal, the resultant in the
case of the same or opposite phases will be in a straight line, the
diagonal of a rectangle, not of a square. With the difference of
phase of J period, the circle becomes an ellipse with its axes coinci-
dent with the directions of the component motions.

25. Composition of S. H. M.''s of different periods. — The motion
resulting from the composition of two S. H. M.'s of different periods
is more or less complicated according to the ratio of the periods.
We have already discussed the case where the ratio is 1:1. The
next simpler ratio is 1 :2.

KINEMATICS.

Experiment 2. — Suspend from two tack staples in a horizontal
bar a loop of string (say) 2 m long. Slip the end of the loop through
a small clamp which may be adjusted at any height (Fig. 20).
|] Adjust so that P = i C P. Cause the
B pendulums OP and CP to oscillate in
planes at right angles to each other. As
CP is 4 OP, the period of CP is twice
that of OP (p. 78). By following the
directions given in the last experiment,
the resulting motions may be observed
and studied. Fig. 21 represents the path
of P when the phase differences are re-
spectively 0, i, i and f period.

Should the ratio be not exactly 1:2,
the different curves will gradually change
from one to another. The result of com-
pounding S. H. M.'s of other period ratios
may be studied in a similar manner by
slippiflg the clamp up or down and thus
changing the relative lengths of the two
Fig. 20. pendulums. The figures described on the

paper by the falling sand are exceedingly intricate and interesting.

Fig. 21.

26. Composition of S. H. M. with a uniform motion at right
angles to it ; harmonic curve. — Let the S. H. M. of a particle
be executed in the line AB (Fig. 22); and let the particle at
the same time travel with uniform speed from left to right.
ACB is the circle of reference, which is divided into fourteen
parts of equal length by the points AC DE, etc. Lines drawn
through these points at right angles to the line AB determine
the points on this line (viz., c, d, e, etc.) which define the
positions of the particle, so far as determined by the S. H. M.,

WAVE MOTIONS.

at equal intervals of J^ of the period. Lines drawn parallel
to AB divide the space into equal intervals, which represent
the distances traversed by the particle from left to right
during each ^l of a S. H. M. period. Now combining the
motion AM with Ac, MN with cd, NP with de, etc., we get
points 1, 2, 3, etc., which represent the actual positions of
the particle after successive intervals of yL of a period. If
these points be joined by a gently curving line, there results
a characteristic curve called a Harmonic Curve.

A M N P

~Y^

r^r

. &

jS^

/'-/

W-

—

Fig. 22.

Experiment 3. —Partly fill the funnel pendulum (Fig. 19) with fine
sand and suspend it from a frame. Set it swinging like a pendulum bob
and beneath it move uniformly a sheet of paper at right angles to the
plane in which the pendulum swings. The falling sand will be deposited
in a curve which is approximately a harmonic curve.

The movement of the pendulum is approximately S. H. M.,
hence the harmonic curve is the resultant of the S. H. M. of
the funnel and the uniform motion of the strip of paper.

27. Wave 7notio7is. — A, B, C, D, etc. (Eig. 23) represent a
series of particles lying in the same straight line, e.g. a series

M N

DD3QO

Fig. 23.

of particles of water lying in the smooth surface of a body of
water. Just below, the same particles are represented as
moving simultaneously, each in a -circular path, in a vertical

32 KINEMATICS.

plane. Particle B is just i of a period behind A, C the same
interval behind B, and so on. A line drawn through the par-
ticles in their several positions at the same instant is called
a wave liiie. As long as the particles continue to move in
their respective circles, so long will a ivave form traverse the
series of particles. If a person is favorably placed so that
he can observe a series of water waves passing him, he will
perceive that floating blocks of wood move in elliptical paths
(the circular form is not a necessary attribute of these
motions), never moving more than a certain distance from
certain points about which they oscillate. The motions of
the blocks represent rather imperfectly the curvilinear paths
in which the particles composing a body of water move while
that body is agitated by waves. To the observer, ridges and
furrows of water appear to move along the expanse, but
objects floating upon the surface are not carried along by
them, which shows that the appearance is a deception, and
that the body of water is traversed only by wave forms.
Observe that, whereas in Fig. 22 the harmonic curve results
in a combination of an harmonic and a uniform translatory
motion, in Fig. 23 the wave line results from a transmission
of circular motion to a series of particles in such a manner
that the motion of any particle shall be a definite part of a
period behind that of its predecessor. Particles B and J are
in positions of maximum displacement in the same direction,
and particles B and F are in positions of maximum displace-
ment in opposite directions. The distance from B to J, or
the distance from any particle to the next particle which is
in the same relative position in its movement, is called a wave
length; the distance 5i^is a half ivave-length.

DYNAMICS.

PART I. — CHAPTER I.
MOLAR DYNAMICS.

Section I.

FORCE. MOMENTUM.

28. Dynamics is the science which treats of the action
of force. This science will be treated under three heads :
(1) Molar Dynamics 5 that is, the dynamics of solids and
fluids, including the study of sound waves ; (2) Molecular
Dynamics, including heat ; (3) Ether Dynamics ; that is,
radiation, including light and electricity.

For present purposes at least, we may regard the term
Physics^ merely as a generic term which includes all these
branches. Hence, Physics is the science which treats of the
dynamics of masses, molecules, and the ether.

29. Force. — When a body at rest is set in motion, or one
which is in motion is accelerated (positively or negatively),
or when a moving body is deflected from a straight course,
experience teaches us that there is always a cause, and we
have also learned to apply to this cause the name force. We
have also learned that when a body is under the influence of
a force which tends to cause a change of motion in that body,
another force must act on the same body if a change of motion
is to be prevented.

We get our primitive idea of force from the sense of
muscular exertion which we experience when, by personal
effort, we put bodies in motion, or stop bodies that are in
motion. We transfer this conception by analogy to a change

1 Physics is often defined as the Science of Matter and Energy, since " In the
physical universe only matter and energy exist independently of our senses and
reason." (Tait.)

34 ,MOLAK DYNAMICS.

of motion observed in any body, and attribute this change to
an interaction between that body and some other body, animate
or inanimate. This interaction is always a jpull or a push and
is accordingly called an attractive or a repellent force. It is
evident that there can be no pull or push except between at
least two bodies or two parts of the same body, i.e., there is
no such thing as a one-sided pull. In other words, when
there is a pull or a push there are at least two bodies pulled
or pushed, and it is only for the sake of convenience in speech
that we are permitted to say that one body pulls and the other
is pulled.

It is not possible for a person to pull without being himself
pulled, or to push without being himself pushed. Appearances
sometimes seem to contradict the above statements. For
example, a man standing on a wharf pulls a distant boat by
means of a rope. The boat moves as the result of the pull,
but, though he is bracing himself against the wharf, he is not
willing, perhaps, to concede that he is likewise pulled. Let
him stand in the boat and pull the rope which is attached at
the other end to the wharf ; both he and the boat move.
What body, according to appearances, is pulled in this case ?
What bodies are actually pulled ?

We are now prepared for a definition of force. Force is an
interaction between two bodies (or two parts of the same body)
causing or tending to cause a change in the motion of each, either
in direction or in magnitude ; or, more simply, force is that
which tends to modify motion.

It should be observed that the above conveys no idea of
what the real nature or essence of force is, for of this we are
quite ignorant. Indeed we know of the existence of force only
by its effects; hence an idle force, i.e. a force producing no
effect, is an absurdity.

30. Force not a property of matter. — We speak of force as
exerted by matter, but, strictly speaking, matter does not of

ACTION AND REACTION. 35

itself exert force. Matter must be set in motion or have some
form of energy (see p. 84) conferred upon it before it can
exert force, so that force is merely a manifestation of energy.

31. Action and reaction. — Force is always dual, inasmuch
as it is always oppositely directed upon two bodies. By a
conventionality of speech we say that one of the two bodies
acts upon the other, and the latter reacts upon the former.
Later on it will be shown that the reaction is always equal to
the action.

The wings of a bird act upon the air, giving a certain
portion of it a rearward motion ; the air reacts upon the wings,
giving the bird a forward motion. The bat strikes the ball,
imparting to it an acceleration, the ball reacts upon the bat,
giving it a negative acceleration.

32, Time required for bodies to gain or surrender velocity . —
If a sled on which a child is sitting be suddenly put in motion,
the child is left in the place from which the sled started. If
the child and sled are both in motion, and the sled be suddenly
stopped, the child lands some distance ahead. If the sled be
started slowly, the child partakes of the motion of the sled,
and is carried along with it ; and if the sled gradually stop,
the child's motion is gradually checked, and it retains its
place on the sled. This shows that masses receive motion
gradually and surrender it gradually.

Even very small bodies require time to gain or surrender a definite
velocity. The sand-blast, employed for engraving figures on glass,
furnishes a fine illustration of this fact. A box of fine quartz-sand
is placed in an elevated position. A long tube extends vertically
down from the botton of this box. The plate of glass to be engraved
is covered with a thin layer of melted wax. The design is sketched
with a sharp-pointed instrument in the wax when cool, leaving the
glass exposed only where the lines are traced. The plate is then
placed beneath the orifice of the tube, and exposed to a shower of
sand. The velocity of the sand-grains is not at its maximum at the
start, but is constantly accelerated till thev reach the plate, where^

36 MOLAK DYNAMICS.

in turn, their velocity is gradually given up. The wax, on account
of its yielding nature, gradually brings them to rest ; but the glass,
notwithstanding its hardness, cannot stop them quite at its surface ;
and, therefore, it suffers a chipping action from the sand. Thus the
soft wax affords a protection from the action of the falling sand, for
all parts except those intended to be cut. A still greater force is
generally given to the sand by steam blown through the tube. For
this reason the apparatus is called a sand-blast. Hard metals like
steel are engraved in the same manner. Yet the hand may be held
in the blast several seconds without injury.

Question.

What is the difference in the effects of catching a base-ball with hands
held rigidly extended, and with hands allowed to yield somewhat to the
motion of the ball ?

33. Momentum. — A small stone dropped upon a cake of
ice produces little effect ; a large stone dropped upon the ice
crushes it. An empty car in motion is much more easily
stopped than a loaded car. Every one knows that the effort
to stop a moving body depends to some extent upon the mass
of the bod}^ We have an instinctive dread of the approach
of large masses.

Again, we have a similar dread of masses moving with
great velocities. A ball tossed is a different affair from a ball
thrown. Thus we are led to the consideration of the mass of
a body multiplied by its velocity. This product is called
momentum. A large mass, moving slowly, has great momen-
tum, but the same mass will have twice the momentum if
its velocity be doubled ; again, a small mass, moving swiftly,
has great momentum, but its momentum is increased in pro-
portion as its mass is increased. A unit of momentum is the
momentum of a unit mass moving with unit speed.

If the motion of a mass of 1 k, having a velocity of 1 m
per second, is taken as a unit of momentum, then a mass
of 5 k, moving with the same velocity, would have a

"cvrpuLSE. 37

momentum of 5 ; and if the latter mass slionld have a velocity
of 10 in per second, its momentum would be 5 X 10 = 50.
Hence, the iiumeric of momentum is found by imdtiplying units
of mass hy units of velocity ; in other words, the product, MV,
of a mass, 31, by its velocity, V, is its momentum.

If the mass and the velocity both be unity, the momentum
will also be unity ; and the unit of momentum, which has
received no special name, may be defined as the momentum
of a unit mass moving with unit velocity, and momentum
may be defined as rate of mass-displacement.

The dimensional of the momentum of a body is the product of the
dimensionals of its mass [M] and its velocity [LT-i], i.e. [MLT-i].

Since momentum is a quantity which has direction,
momentum may be compounded and resolved like motions and
velocities.

By experiment, we learn that a given force acting for two
units of time produces twice the velocity that it does in one
unit of time, and that the velocity which a given force produces
is proportional to the time it acts ; hence momentum, when
the force is constant, is equal to the product of the force and
the time, i.e. Ft\ oi MV= Ft. If the force be not constant,
then the momentum must be computed from the average force
acting. The product M V signifies that the mass-motion, or
momentum, of a body depends on its mass as well as its velocity.
The product Ft signifies that the momentum imparted to a
body depends upon the time (t) during which a force acts
as well as upon the intensity (F) of the force. We infer from
the above equation that a definite force acting upon any mass
for a given time will generate in it a speed whose magnitude
is inversely as the mass.

34. Impulse. — The product of the time during which a
force acts by its mean intensity is called the impulse of the
force. This term is usually restricted to a force acting for a

38 MOLAB DYNAMICS.

short time, as in a blow given to a ball by a bat. There is no
propriety in asking : " With what force does the bat hit the
ball ? " The inquiry may be with reference to the average
pressure which it exerts on the ball, or, more likelj^, with
reference to the impulse of the force, i.e. the product of
the mean intensity and the time it acts. It has just been
shown that Ft^MV, hence we infer that the impulse of a
force is measured by the momentum produced.

Questions and Problems.

1. What agent is the immediate cause of motion ?

2. What distinction do you make between velocity and momentum ?

3. Upon what does the momentum given to a ball fired from a gun
by the expanding gases depend ?

4. Inasmuch as equal forces are exerted for the same length of time
by the gases on the ball and the gun, how will the momenta communicated
to each compare ?

5. If there be 25 lbs. of matter in the gun and 1 oz. (yL ib.) in the ball,
and the gun acquire a maximum velocity of 3 feet per second, what, at
that instant, is the velocity of the ball ?

6. Can any body be put in motion in no time ? (Demonstrate from
ioTmnlsiFt= MV.)

7. Compare the momentum of a car weighing 50 tons, moving 10 feet
per minute, with that of a lump of ice weighing 5 cwt., at the end of the
third second of its fall.

8. With what velocity must a boy weighing 25 K move to have the
same momentum that a man weighing 80 K has when running at the rate
of 10 Km per hour ?

9. Since Ft = M V, to what is change of momentum proportional ?

10. If the same force act for the same length of time upon bodies hav-
ing different masses, to what will the velocities produced be proportional ?

11. Two boats of unequal masses are brought together by pulling on a
rope. a. Resistance being disregarded, how will their momenta at any
given instant compare ? 6. How will their velocities at the same instants
compare ?

12. If the motion of the moon in its orbit were to cease, these bodies
would approach each other. The mass of the earth is about 80 times that
of the moon. What part of the whole distance between them would
the moon move before collision ?

FORCE OF GRAVITATION, WEIGHT. 39

35. Force of gravitation^ iveight. — It has been said that
the best way of defining gravitation is to ''let a stone
drop." In this phenomenon we discover evidence of the
action of ^a force between masses ; this force is called gravita-
tion, or gravity when the action is between the earth and some
other mass. Inasmuch as by the action of this force the
earth and stone are brought together^ this force has been
assumed to be, and is universally spoken of as, an attractive
force ; but the probability is that the earth and stone do
not draw each other together, but are brought together
through the agency of some surrounding medium, and the
action is quite as likely to be a push as a pull. The term
weight signiiies the magnitude of the force of gravity which
exists between any body and the earth. It is usually
determined by measuring the pressure which gravity causes
the body weighed to exert upon a supporting body, e.g. on a
scale pan, or the distortion Avhich it produces in the supporting
body, e.g. the elongation of the spring in the spring-balance.
The units generally employed are the 2^ound and kilogram,
and are called the gravitation units of force. All forces may
be measured in the same units. To say that a man pulls a
boat with a force of one hundred pounds is equivalent to
saying that he pulls with a force that is equal to the force
which acts between the earth and a body having a mass of
one hundred pounds. A force of one pound, then, is an
abbreviated expression for a force equal to the weight (at the
locality in question) of one pound of matter.

Sectiox II.

MEASUREMENT OF FORCE.

36. Force tends to produce acceleration. — Thus the force of
gravity causes bodies to fall with accelerated velocity ; it
also transforms the otherwise constant velocity of a body

40 MOLAR DYNAMICS.

projected upward into a retarded velocity. A constant force
acting upon a free body (i.e. a body which encounters no
resistances) always 'produces a uniformly accelerated motion.
This is best illustrated by the fall or ascent of a^body in a
vacuum, the body being meantime acted on only by the
constant force of gravity.

37. Absolute measurement of force by direct observation of
acceleration and mass. — If a force F be applied to a certain
mass m for a unit of time, a certain momentum is generated
in the mass. If the same force be applied to a greater mass
for the same time, it will move with as many times less
velocity as the mass is times greater, but the product of the
mass and the velocity, i.e. the momentum, is the same. That
is, the same force acting for the same length of time on free
bodies having different masses may be measured by the change
of momentum generated by it in a unit of time {e.g. a second),
since this is constant and depends on nothing but the force.
That is, F^=ma, in which F represents any constant force
acting on any mass m, a the acceleration, and m a the rate of
change of momentum. Force is sometimes defined as the time
rate of change of m.or)ientum.

38. Absolute units of force. — A unit of force in the absolute
system is that which, acting for a unit of time, will give to a
unit of mass a unit of acceleration. The absolute unit of
force (in the C. G. S. system) is called a dyne, and is that
force ivhich in one second is capable of giving to a gram-mass
an acceleration of one centimeter per second ; in other words, it
is a constant force of the requisite intensity to impart in one
second to a gram-mass a velocity of one centimeter per second.

Any constant force which in one second produces in a mass
of m grams an acceleration of a centimeters per second must
be equal to m X a dynes {i.e. F=^m. a'). In physics the letter g
is generally used instead of the letter a to denote the accelera-
tion due to the force of gravity. By exact measurement the

EXPRESSION FOR THE MASS OF A BODY. 41

acceleration produced by the force of gravity on free bodies
(i.e. in a vacuum) is found to be, in the latitude of Boston at
the level of the sea, 980.4 cm per second. Hence the force of
gravity acting on a mass of one gram must be (substituting,
in the equation above, W (weight) for F, and g for a)
W=7ng = l X 980.4 = 980.4 dynes. ^ Consequently it requires
a force of 980.4 dynes to support (i.e. prevent from falling)
a mass of one gram ; or the weight of a gram-mass at sea level
in latitude 42° is 980.4 dynes. A dyne^ is therefore about -^^^
of the weight of a gram-mass, or exactly 9 g^.e ^f the weight of
a gram-mass at Paris. In the gravitation system the weight
of a gram-mass is a gram-force, hence 1 gram-force = 980.9
dynes at Paris. Gravitation units in grams-force at Paris are
readily changed into dynes by multiplying by 980.9 ; at
Boston, by multiplying by 980.4 ; and generally by multiplying
by the value of g at any place.

From the foregoing statements it appears that the weight
of the gram-mass varies with locality, e.g. Boston or Paris,
so that the value of a gravitation unit of force, e.g. a gram-
force, is variable. This want of definiteness constitutes
a serious objection to the gravitation system when great
accuracy is required. The value of the dyne is definite,
absolute.

39. Exjjression for the mass of a body in terms of its weight. —

Since W=7ng, m^ — ; that is, mass is measured by its

weight in poundals ^ or dynes, divided by the acceleration in feet
or centimeters per second produced by gravity. Although W

1 The equation W=:'mg expresses the fact that the number of dynes (of gravity)
acting on a given mass is g times the number of grams-mass in that body. A similar
statement holds for other systems of units.

2 A dyne is a very small force. In expressing a force of considerable magnitude
the megadyne (a million dynes) is commonly used.

3 A poundal is that force which in one second is capable of giving to a pound-mass
an acceleration of one foot per second.

42 MOLAR DYNAMICS.

and g vary with latitude and elevation^ tliey vary proportion-

W
ally, hence the ratio — {i.e. the mass) does not change, but

is constant for the same body.

A body suspended from a spring balance (see Fig. 24) is found to weigh
at Paris \ K. Required its weight in dynes. Solution : i K = 500 g :
500 X 980.9 = 490,450 dynes.

Required to find the force which, acting for 10 sec. , gave to a mass of

10 g a velocity of 1000 cm per sec. Solution : F=: ^ = 10 X — — - = 1000

dynes.

Required to find the mass in which a force of 1500 dynes produces an

acceleration of 2 cm per sec. Solution: m =— = -^— — 750 g.
^ a 2 ^

Required to find the acceleration which a force of 2000 dynes can give

a mass of 4 g. Solution -. a — — — — ^— = 500 cm per second.

m 4

40. Measurement of force hy countei'halancing . — The house-
hold instrument called a spring balance is strictly speaking a
dynamometer, i.e. a force-measurer. It contains a
spiral spring as seen in A (Fig. 24) carrying an
index which moves over a scale as shown in B.
If a unit of mass (e.g. 1 lb. or 1 K) be hung upon
the spring, it is lengthened by a certain definite
quantity. If, grasping the ring in one hand and
the hook in the other, you lengthen the spring by
a muscular pull as much as it was lengthened by
the force of gravity acting on the mass, the infer-
^^' ^' ence is that the muscular force which you exert
is equal to the force of gravity exerted on the mass, hence the
spring balance measures all forces in gravitation units. A
spring balance might, however, be graduated in dynes so as to
measure force in absolute units.

The pound and the gram are primarily units of mass. A
pound-force is a force equal to the weight of one pound of matter.

TWO SYSTEMS OF MEASUREMENT OF FORCE. 43

41. Two systems of measurement of force. — We have found
in tlie foregoing discussions that there are two methods of
measuring force : one specially adapted to measuring balanced
forces (see p. 45), called the statical or gravitation system ; the
other specially adapted to measuring unbalanced forces (see
p. 46), called the kinetic or absolute^ system ; though a force,
whether balanced or unbalanced, may always be measured by
either system. The gravitation system is so called because, by
it, forces are compared with the force of gravity as a standard.
The two methods of measuring force give rise to tiuo syster}is
of units called respectively the gravitation and the ahsohite
systems, either one of which is easily convertible into the
other, as shown above. ^

Questions and Problems.

1. A constant force acts on an otherwise freely moving body in a
direction opposite to that in which it is moving ; how is tlie body's motion
affected thereby ? Give an illustration.

2. How does a gravitation unit of force differ from an absolute unit ?

3. Would a spring balance graduated in grams in Paris and sensitive
to the smallest changes answer for weighing {i. e. determining the exact
masses of bodies) in Boston ?

4. What kind of a motion does a constant force produce on a free
body ? How has this been shown ?

5. To what is the acceleration produced in equal masses proportional,
ie. if m is constant, a will vary as what ?

6. On what condition will equal forces produce equal accelerations ?

7. Suppose that you fill a box with sand, place it on a toy cart, pull
the cart by a string with a constant force along a smooth floor for a
certain number of seconds, and observe the acceleration given the load
(cart, box, and sand), then remove the sand and replace it with lead shot;
how could you tell, by pulling the load with the same force as before,
when it has the same" mass as the former load ?

1 Measurements of force in the absolute system are attended with serious practical
difficulties in the way of observation of the acceleration produced, yet the absolute
units are almost indispensable in very many scientific calculations, especially in
electricity and magnetism.

44 MOLAR DYNAMICS.

8. a. Has the same mass equal weights in Paris and Boston ? b. How
sensitive must a spring balance be to discover any difference ?

9. Show that a spring balance is, strictly speaking, a force-measurer,
and not a mass-measurer.

10. a. When we speak of a force of one pound, what do we mean ?
6. When we speak of a force of one dyne, what do we mean ? c. When
we speak of a mass of one pound, what do we mean ?

11. a. If one mass is four times another, how many times as much
force is necessary to produce the same acceleration in the former as in
the latter ? b. How many times greater is the force of gravity acting on
a mass of one hundred pounds than on a mass of one pound ? c. If a
hundred-pound iron ball and a one-pound iron ball be let drop from the
same hight at the same instant, which ought to reach the ground
first?

12. A body weighing 4 g is moving with an acceleration of 12 cm per
second ; what is the force acting ?

. 13. A body acted on by a force of 100 dynes receives an acceleration
of 20 cm per second ; what is its mass ?

14. A body of m.ass 30 g is moved by a constant force of 50 dynes ;
what is its acceleration ?

15. What force acting on unit mass for unit time.will cause it to move
with unit velocity ?

16. What acceleration will a force of 20 dynes produce on a mass
of lOg?

. 17. What velocity will a force of 20 dynes acting on 1 K impart to it
in 5 minutes ?

18. a. What is the weight in dynes of a mass of 1 K in Boston ?
b. How many more dynes does it weigh in Paris ?

19. A constant force of 20 dynes acts on a mass of 5 g and gives it a
velocity of 500 cm per second ; how many seconds does it act ?

20. How is the value in dynes of a gram weight at any locality deter-
mined ?

21. How much is the momentum of a gram-mass changed by gravity
in one second when falling freely in Boston ?

22. Why will a bullet fired at an open door pass through it without
moving it perceptibly, while a push of the hand, of much less intensity,
moves it an appreciable distance ?

23. Explain why the weight of a body is not a perfect measure of its
mass.

24. What is the relation of the static unit of force to the kinetic
unit?

GRAPHICAL BEPRESENTATION OF FOKCE. 45

Section III.

COMPOSITION AND RESOLUTION OF FORCES.

42. Grajpliical representation of force. — A force is defined
when its magnitude, direction, and 'point of application, are
given. Hence we may represent forces graphically by straight
lines whose lengths bear to one another the same relation as
the numerics of the forces, while the directions of these lines
indicate the directions of the forces, and the points from
which the lines are drawn indicate the points of application.
Thus, on a scale of 1 cm =: 1 k the
line A B (Fig. 25) represents a force
of 3.2 k acting toward the right with ^ ^ ^

its point of application at A ; and ^^^' ^'

the line D E represents a force of 2 k acting parallel to the
first with its point of application at D.

43. Composition of forces acting hi the same line ; equilibiHum
of forces; balanced and unbalanced forces.

Experiment. — Insert two stout screw-eyes in opposite extremities of
a block of wood. Attach a spring balance to each eye. Let two persons
pull on the spring balances at the same time, and with equal force,
as shown by the indexes, but in opposite directions, — The block does
not move. One force just neutralizes the other, and the result, so far as
any movement of the block is concerned, is the same as if no force acted
on it.

When one force opposes in any degree another force, each
is spoken of as a resistance to the other. Let / represent the
number of pounds of any given force, and let a force acting
in any given direction be called positive, and indicated by the
plus (+) sign, and a force when acting in an opposite direc-
tion to the force which we have denominated positive, be called
negative, and indicated by the minus ( — ) sign. Then if two
forces +/ and — / acting on a body at the same point or along

46 MOLAR DYNAMICS.

the same line are equal, they are said to be balanced, and the
result is that no change of motion is produced.

Viewed algebraically, +/ — /^ ; or, correctly interpreted,
-\-f—f=z= (is equivalent to) 0, i.e. no force. In all such
cases there is said to be an equilibrium of forces, and the body
is said to be in a state of equilihriwn.

A force that produces equilibrium with one or more forces
is called an equilibrant.

If one of the forces be greater than the other, the excess is
spoken of as an unbalanced force, and its direction is indicated
by one or the other sign, as the case may be. Thus, if
a force of -f- 8 pounds act on a body toward the east, and
a force of — 10 pounds act on the same body along the same
line, then the unbalanced force is — 2 pounds ; i.e. the
result is the same as if a single force of 2 pounds acted on
the body toward the west. Such an equivalent force is called
a resultant. A resultant force is a single force that may be
substituted for ttuo or more forces and produce the same result
that the simultaneous action of the several forces would produce.

The resultant of any number of forces acting m the same
straight line is equal to the algebraic sum of the forces. An
equilibrant of several forces is equal in magnitude to their
resultant, but opposite i7i direction. The process of combining
several forces so as to find their resultant is called composition
of forces. The forces combined are called the components.
The converse operation, of finding component forces which
shall have the same effect as a given force, is called resolution
of forces.

44. An unbalanced force alivays pjroduces acceleration. —
A body acted on by an unbalanced force cannot be at rest.
That branch of dynamics which treats of the relation of force
to the motion which it produces is called kinetics, and that
branch which treats of equilibrium of forces is called statics.

Equilibrium is often maintained by the reaction of a sur-

PRESSURE, TENSION. 47

face with which the body acted on is in contact. A simple
illustration is that of a body supported on a horizontal sur-
face, as of a table. Here the reaction caused by the com-
pression of the material of which the table is composed is
equal to the weight of the body.

45. Pressujx, tension. — A balanced force does not produce
acceleration, but causes either a pressure or a tension. A force
exerts pressure when it tends to compress or shorten in the
direction of its action the body on which it acts. Examples :
Pressure exerted on the springs of a carriage, on air when it is
compressed in an air gun, etc. A force causes tension when it
tends to lengthen in the direction of its action a body on which
it acts. A body thus subjected to a force tending to elongate
it is said to be in a state of tension., and the stress to which it
is subjected is called its tension, and the strength to resist
being pulled apart which it possesses is called its tensile
strength.

Questions and Exercises.

1. Explain the use of a line to represent force ?

2. a. When a force of 100 lbs. is represented by a line 5 inches long,
what is the scale ? 6. "What force will a line J in. long represent on the
same scale ?

3. a. Represent on a scale of i inch = 1 lb. the resultant of forces of
5 lbs. and 7 lbs. acting in the same direction. (Always place arrow
heads in lines representing forces to indicate the direction of the forces.)
b. Show, by points A, B, and C placed in the line, the components of this
resultant, c. Represent the same two forces acting in opposite directions
upon the same point A. d. How will you represent the resultant of
these two opposing forces ?

4. Three men, A, B, and C, pull on a rope in the same direction with
forces respectively of 50 lbs. , 60 lbs. , and 70 lbs. A is nearest the end
of the rope, B next, and C next. a. What is the tension of the rope
between A and B ? 6. What, between B and C ? c. A man, D, just
beyond C pulls with a force of 75 lbs. .in the opposite direction. With
what force must a man, E, pull, that there may be equilibrium ?
d. When there is equilibrium, what is the tension of the rope between

48 MOLAR DYNAMICS.

C and D ? e. How great must be the tensile strength of the rope
between C and D ? /. Write the equation showing the algebraic ad-
dition of the forces in case of equilibrium.

Section IV.

COMPOSITION OF PARALLEL FORCES. MOMENTS OF FORCES.

46. Composition of ijarallel forces acting in the same
direction and in the same ^jZa?ze.

Experiment. — AB (Fig. 26)
represents a rod lying on a table
with three strings loosely looped
around it so that they may be
slid along the rod. Dynamom-
eters are attached to the free
ends of the strings. The strings
are all stretched in parallel direc-
tions in a plane parallel with the
top of the table. (Great care must

Fig. 26.

be taken in the manipulation to keep the three strings exactly parallel.)
The dynamometers register the tensions in the several strings, i.e. the
forces applied through them to the rod.

Observe (I) when there is equilibrium the dynamometer
E registers as much as those of F and G added together.
(This would be true if more than two forces were applied in
the same direction as A F and B Gr.) But the force applied at
C is the equilibrant of the other forces and this is equal to
their resultant acting in the direction CD. (II) The point
of application of the resultant (or equilibrant) is between the
points of application of the components. (HI) This point is
nearer the greater force. (IV) The distance of this point
from the smaller force is as many times greater than its
distance from the larger force as the larger force is times the
smaller force. For example, if A F be 14 lbs. and B G 6 lbs.,

UNEQUAL PARALLEL FORCES. 49

i.e. 14 : 6 — 7 : 3, then distances C A and B will be as 3 : 7.
In other words the component forces are said to vary inversely
as,^ or to be inversely proportional to^ their distances from their
resultant. These observations are summarized as follows :
The resultant of tiuo parallel forces in the same direction is
equal to their sum, and the distances of their points of apjplica-
tionfrom the point of application of the resultant vary iyiversely
as the intensities of the components.

Corollary : The condition of eq_uilibriuin is that the algebraic
sum of the forces (positive and negative) must he zero.

When more than two forces act on a body in the same
plane and in the same direction, the resultant of any two of
them (and its point of application) is found, then the resultant
of this resultant and a third force, and so on until all have
been used.

47. Composition of two unequal parallel forces acting in
opposite directions. — Let F and F^ (Fig. 27) be parallel forces
acting in opposite directions on C B, of which F is the greater.
The force F may be resolved into two
forces : one, represented by BBi, equal
and opposite to Fi ; the other equal to
F — Fi, and represented by the line C D.
But the forces F^ and B Bi are in equi-
librium, leaving an unbalanced force at
C equal to F — Fj. This, then, is the
resultant E of the forces F and Fi, i.e.
R=:F— Fi. But BBi:CD=AC:AB, ^ ^

and by composition B Bi + C D : B Bi= ^"^^^-^t-

AC + AB:ACorF:Fi = CB:CA. Hence the resultant of two
unequal parallel forces acting in opposite directions is equal to
their difference, and acts outside of both in the direction of the

1 The pupil should acquire immediate familiarity with these expressions of fre-
quent occurrence in physics, and should practice in this connection writing inverse
proportions. Thus for the quantities here given, 14 : 6 — J : i, i.e. the forces are
proportional to the reciprocals of their respective distances from the resultant.

50 MOLAR DYNAMICS.

greater component, and the distances of its point of appli-
cation from the points of application of the two forces are
inversely proportional to their intensities.

48. Dynamical couple. — If F and Fi (Fig. 27) be equal, the
magnitude of the resultant, being equal to the difference of the
components, is zero, i.e. they have no resultant. Two equal
forces applied to the same body in parallel and opposite directions
not in the same line constitute what is called a ^^ couple.'^
The effect of a couple is to produce rotation, but no motion
of translation. The value of a couple will be determined
later on.

49. Moment of a force. — The value of a force to produce
rotation around a given axis is called its moment with refer-

A 3ft. c 3ft. B ^^^^ to that axis. The

ZS axis is, of course, al-

Ibs. 20 lbs. V , • 1 ,

ways a line at right

D ^ angles to the plane

^^^•28- of rotation. Point C

(Fig. 28) may represent the extremity of the axis about which

A B is supposed to rotate. The perpendicular distance (C A

or C B) from the axis of rotation to the line of direction in

which a force acts (A D or B E) is called the arm or leverage

of the force.

The Tnoment of a force is measured by the product of the
intensity of the force into the
arm. For example, the moment / ^"\ '' " "^

of the force AD (Fig. 2^) is A/-----_\c/y^^^\,^^^
expressed numerically by the / ^^ ^^Q

number (30 X 2 =) 60, and the /
moment of B E is (20 X 3 =) 60. >
BydefinitionthelineAC(Fig.29) ^'''- ^^•

is the arm of force P, and B C of the force Q.

50. Equilibrium of moments. — The moment of a force is
said to be positive when it tends to produce right-hand rota-

MOMENT OF A COUPLE.

tion, i.e. in the direction in which the hands of a clock move,
and negative when its tendency is in the reverse direction.
If two forces act at different points of a body which is free to
rotate about a fixed pointy they will produce equilihriuifn when
the algebraic sum of their moments is zero. Thus the moment
of the force applied at A (Fig. 28) is — (30 X 2) = — 60.
The moment of the force applied at B in an opposite direction
is accordingly + (20 X 3) = + 60. Their algebraic sum is
zero, consequently there is equilibrium between the moments,
and no tendency to rotation.

When more than two forces act in this manner, there will
be equilibrium if the sum of all the positive moments be
equal to the sum of all
the negative moments.
Thus, the sum of the
positive moments act-
ing about point D (Fig.
30) is (/) 45 + (e) 25
+ (a) 30 = 100 ; the
sum of the negative moments acting about the same point is
(c) 30 + {d) 40 + {b) 30 = 100 ; the two sums being equal, the
moments are in equilibrium.

51. Moment of a coupjle. — The moment of a couple, or its
Pi value in producing rotation, is the sum of
the moments of its two components around
the axis of rotation. Let F and Fi consti-
tute a couple whose arm is AB (Fig. 31).
To find the rotating value of the couple, let
P be the axis of rotation, then the moments
of F and Fi relatively to P are FxAP,
and Fi X B P. The total resultant moment
of the two forces is (F X AP) + (Fi XB P), or (since F^Fi)
FXAB.

1>6

5 D

xo ''■'

iy2

Fig. 30.

Fig. 31.

52 MOLAR DYNAMICS.

Questions and Problems.

1. Two parallel forces of 8 lbs. and 12 lbs. act in the same direction
respectively at points A and B, 12 inches apart. Find the magnitude
and position of their resultant.

2. ■ The smaller of two parallel forces having the same direction is 5
inches from the resultant ; what is the distance of the resultant from the
other force ?

3. Two men carry a weight of 100 lbs. suspended from a pole 15 feet
long ; each man is 18 inches from his end of the pole. Where must the
weight be attached in order that one man may bear f of it ?

4. Take from the last problem the number of pounds supported by
each man and the respective distances of each from the weight, and make
an inverse proportion which shows the relation that must exist between
these quantities.

5. How can a force of 4 lbs. be made to produce equilibrium with a
force of 12 lbs. ?

6. Draw a line 2 inches long. Kepresent on a scale of i inch = 1 lb.
a force of 8 lbs. applied at a point A :^ of 1 inch from one end of the line
and at right angles to it. Take for the axis of rotation a point B f inch
from the same end of the line. From point C i inch from the other end
of the line draw a line which will represent a force that will produce
equilibrium with the first force, and thereby prevent rotation.

7. Repeat the work of the last problem except that the force applied
at A shall act obliquely on the line.

8. Can a single force produce equilibrium with a couple ?

9. a. A plank weighing 40 lbs. is placed across a log so as to be
balanced. A boy weighing 60 lbs. sits on one end of the plank. Where
shall another boy weighing 90 lbs. sit that he may balance the first?
b. What pressure will be exerted upon the log ?

10. Two horses harnessed abreast are ploughing. How can you
arrange that one horse shall pull only two-thirds as much as the
other ?

11. The maximum muscular force which a certain man can exert is
200 lbs. With what leverages can he raise a stone weighing a ton ?

12. How can pressure be multiplied indefinitely ?

CENTEK OF MASS DEFINED.

Section V.

CENTER OF MASS OR CENTROID.

52. Center of mass defined. — Let Fig. 32 represent any
body of matter ; for instance, a stone. Every particle of the
body is acted upon by the force of gravitation. The forces of
gravitation of all the particles form a set
of parallel forces acting vertically down-
ward, the resultant of which equals their
sum (§ 43), and has the same direction as its
components. The resultant passes through
a definite point in whatever position the
body may be, and this point is called its
center of mass, or centroid. The center
of mass (cm.) of a body is, therefore, the
point of application of the resultant of
all these forces ; and for practical purposes the whole Tnass of
the body may be supposed to be concentrated at this point. ^ By
the place or location of a body mathematicians mean that
point where its center of mass is situated.

It is evident that in whatever position a body be placed,
the resultant of the lueights of all its particles passes through its
centroid. Hence, to support a body (i.e. to prevent its falling),
the supporting force, or equilibrant, — or the resultant of several
supporting forces, — mMst act in a line through the centroid of
the body and vertically upward. A vertical line is any straight
line passing through the centroid of the earth. Up and down
are directions in this line from and toward the earth's centroid.

Let G in the figure represent the cm. of the stone. For
practical purposes, then, we may consider that the force of

^. ^ The expression center of mass does not necessarily signify that point occupying

■ * a central position among the particles of a body, but a point where, for convenience

in dynamical problems, we may consider all the mass (or inertia) to be concentrated.

54 MOLAR DYNAMICS.

gravitation acts only at this point, and in the direction GF.
If the stone fall freely, this point cannot deviate from a
vertical path, however mnch other points of the body may
rotate about this point during its fall. Inasmuch, then, as the
cm. of a falling body always describes a definite path, a line
GF that represents this path, or the path in which a body
supported tends to move, is called the line of direction. It
may be defined as a straight line in which lie the centroid
of the body and the centroid of the earth.

To support any body, then, it is only necessary to provide a
support for its centroid. The supporting force must be applied
someivhere in the line of direction. The difficulty of poising a
book, or any other object, on the end of a finger, consists in
keeping the support under its centroid, i.e. in the line of direction.

Fig. 33 represents a toy called a " witch," consisting of a cylinder
of pith terminating in a hemisphere of lead. The toy will not lie in
a horizontal position, as shown in the
figure, because the support is not
applied immediately under its cm. at G-\
G ; but when placed horizontally it
immediately assumes a vertical po-
sition. It appears to the observer to

rise ; but, regarded in a technical sense, it really falls, because its
cm., where all the mass is supposed to be concentrated, takes a
lower position.

Whether a body having no other support than that applied at
its base ivill stand or fall depends up)on whether or not its line
of direction falls luithin its base.' The .base of a body is not
necessarily limited to that part of the under surface of a body
that touches its support. For example, place a string around
the four legs of a table close to the floor : the rectangular
figure bounded by the string is the base of the table. (What is
the base of a man when standing on one foot ? on two feet ?)

The centroid of any symmetrical body of homogeneous
material (i.e. of uniform density) coincides with its geo-

li^^HB

HOW TO FIND THE CENTER OF MASS OF A BODY. 33

metrical center. Examples : the middle point of a material
straight line ; that point on a straight line joining the vertex
to the middle of the base of a triangle situated at a distance
from the vertex eqnal to two-thirds the length of the line ; the
geometrical center of any polygon, a sphere, a circular cylinder.

53. Hoiv to find the center of mass of a body. — Imagine
a string to be attached to a potato by means of a tack, as in
Fig. 34, and to be suspended from
the hand. When the potato is at
rest, there is an equilibrium of forces,
and the cm. must be Somewhere in
the line of direction an ; hence, if a
knitting-needle is thrust vertically
through the potato from a, so as to
represent a continuation of the verti-
cal line oa, the cm. must lie some-
where in the path an made by the
needle. Suspend the potato from
some other point, as h, and a needle thrust vertically through
the potato from h will also pass through the cm. Since the
cm. lies in both the lines an and hs, it must be at c, their
point of intersection. It will be found that, from whatever
point the potato is supported, the point c will always be verti-
cally under the point of support. On the same principle the
cm. of any body is found. But the cm. of a body may not be
coincident with any particle of the body; for example, the
cm. of a ring, a hollow sphere, etc.

54. Three states of equilibrium. — That a body acted on
solely by the force of gravitation may be in equilibrium, it is
necessary and sufficient that a vertical line through, its centroid
shall pass through the point or surface by which it is sup-
ported. The weight of a body is a force tending downward ;
hence, a body tends to assume a position such that its cm. will
be as low as possible.

Fig. 34.

56 MOLAE, DYNAMICS.

Experiment. — Try to support a ring on the end of a stick, as at b
(Fig. 35). If you can keep the support exactly under the cm. of the ring,
there will be an equilibrium of forces, and the ring will remain at rest.
But if it is slightly disturbed, the equilibrium will be destroyed, and the
ring will fall. Support it at a ; in this position its cm. is as low as pos-
sible, and any disturbance will raise its cm. ; but, in consequence of the
tendency of the cm. to get as low as possible, it will quickly fall back into
its original position.

A body is said to be in stable equilibrium if its position is
such, that any motion except of translation would raise its cm.,
since in that event it would tend to return
to its original position. On the other hand,
a body is said to be in U7istahle equilibrium
when a disturbance would lower its cm.,
since it would not tend to return to its
original position.

A body is said to be in neutral or indiffer-
ent equilibrium when it rests equally well
in any position in which it may be placed.
A sphere of uniform density, resting on a horizontal plane,
is in neutral equilibrium, because its cm. is neither raised nor
lowered by a change of base. Likewise, when the support is
applied at the cm., as when a wheel is supported by an axle,
the body is in neutral equilibrium.

It is evident that if the c.rn. he below the support, as in the
last experiment with the ring, the equilibrium must be stable;
but a body may be in stable equilibrium, though its
cm. be above the point of support. (When is this
possible ?)

It is difficult to balance a lead-pencil on the end of
a finger ; but by attaching two knives to it, as in Eig.
36, the cm. may be brought below the support, and
it may then be rocked to and fro without falling. ^^^- 36.

55. Stability of Bodies. — The ease or difficulty with which
bodies supported at their bases are overturned varies with the

STABILITY OF BODIES.

hight to which their cm. must be raised to overturn them.
The letter c (Fig. 37) marks the position of the cm. of each of
the four bodies A, B, C, and D. If any one of these bodies
be overturned, its cm. must have passed through the arc ci,
and have been raised through the hight ai. By comparing A
with B, and supposing them to be of equal weight, we learn
that of two bodies of equal weight and hight of c.vi., the cm. of
that body ivhich has the larger base viust be raised higher, and
that body is, therefore, overturjied with greater difficulty. A

comparison of A and C, supposing them to be of equal weight,
shows that ivhen two bodies have equal bases and lueights, the
body having its cm. higher is more easily overturned. D and C
have equal masses, bases, and hights, but D is made heavy at
the bottom, and this lowers its cm., and gives it greater stability.

Questions and Exercises.

1. Where is the centroicl of a box ?

2. Why is a pyramid a very stable structure ?

3. What is the object of ballast in a vessel ?

4. State several ways of giving stability to an inkstand.

5. a. In what position would you place a cone on a horizontal plane,
that it may be in stable equilibrium ? b. That it may be in neutral
equilibrium ? c. That it may be in unstable equilibrium ?

6. In loading a wagon, where should the heavy luggage be placed ?
Why ?

MOLAR DYNAMICS.

7. Why are bipeds slower in learning to walk than quadrupeds ?

8. Why is mercury placed in the bulb of a hydrometer ?

9. How will a man rising in a
boat affect its stability ?

10. Which is more liable to be
overturned, a load of hay or a load
of stone of equal weight ?

11. Draw a triangle and find its
center of mass.

12. What attitude does a man
assume when carrying a heavy load
on his back ? Why ?

13. Explain the difference in the
behavior of a ball and of a cube,
when placed on a plane slightly in-
clined.

14. What position do bodies floating in air or in water take ?

15. a. Explain how the toy horse (Fig. 38) stands upon the platform
without falling off. 6. Explain how the toy may rock upon its support
without falling off.

Fig. 38.

Section VI.

COMPOSITION OF FORCES ACTING AT ANGLES WITH ONE
ANOTHER.

bQ>. When the handle A (Fig. 6) is pushed forward, there
is applied to the pencil a force which may be represented in
magnitude and direction by the line a h ; at the same time
the pencil is pulled vertically up by a force which may
properly be represented by the line a c. The pencil, however,
moves in the line a d, which is a diagonal of a parallelogram
constructed on the lines ah and etc. It is evident that a single
force might be applied to the pencil with the same effect that
the two forces produce. Obviously, if a single force were to
move the pencil in the line a d, it must have the direction of
this line. It remains to ascertain whether the diagonal line
ad represents the magnitude of the resultant. Evidently if

COMPOSITION OF FORCES.

this diagonal does represent the resultant, then the same
diagonal with the direction reversed will represent the
equilibrant of these forces. We put the matter to an experi-
mental test with other apparatus :

Experiment. — Insert pegs in any three holes of the circle in the top
of the circular table, Fig. 39. Join these by threads attached to a spring

Fig. 39.

balance as shown in the figure. Stretch the balances so as to indicate
any desired pull in each of the threads. Place under the threads a sheet
of white paper. Locate on the paper the common point of application A
of the three forces. Draw lines AB, AC, and AD, to represent the
directions in which the forces act. Since the point A does not move, it
is evident that the three forces are in equilibrium and that any one of the
three forces is the equilibrant of the other two. Select any one for an
equilibrant {e.g. AD) and extend it in' an opposite direction from A,

MOLAR DYNAMICS.

representing (on some suitable scale) a force A E equal to and opposite to
the force A D as indicated by the dynamometer D, On the same scale
lay off distances A B and A C representing the magnitudes of the forces
acting in the directions of these lines. The line AE is by definition
(§ 43) the resultant of A B and A C. Connect E with C and B. The
figure, if the work be done with care, will be found to be a parallelogram.
The diagonal EA represents the magnitude of the equilibrant of the
forces AB and AC, and the same line with the direction reversed (i.e.
A E) represents the resultant.

57. Parallelogrmn of forces. — If two forces applied at a
point he represented in magnitude and direction hy the adjacent
sides of a paradlelograrn, their resultant luill be represented in
magnitude and direction hy the diagonal which passes through
that point.

This proposition is applicable whether the forces act on a
particle or on a rigid body provided they lie in the same plane.

Thus, let two forces applied at points A and B of a stone
(Fig. 40) act in the directions A C and B D respectively.
The direction of the resultant must pass through E, the point
where the lines of direction of the given forces produced back-

COMPOSITION OF FORCES. 61

wards intersect. If, now, the lines E C and E D be laid off to
represent the relative intensities of the forces, the diagonal
E F of the parallelogram constructed thereon will represent
their resultant, and its point of application may be G or any
other point in the line G H.

58. Composition of more than tiuo forces in the same plane. —
When more than tivo components are given, find the resultant
of any two of them, then of this resultant and a third, and so on
till every component has been used. Thus, in Fig. 41, A C is
the resultant of A B and A D, and __— — ^B

A F is the resultant of A C and A E, A^f^c:::;^'^ ^^ '

i.e. of the three forces A B, AD, / 1^\~~^^~^"^^^~-~~^ ',
andAE. (Invent several problems / \ ^n.-— — ""^^^^
similar to this, in which three, four / d \.

or more forces are to be combined, ~~^~~~-~~.^^ ^s,^^ /

and work out the results.) ~ p

Generally speaking, a motion ^i^- *i-

m.ay he the result of any number of forces. When we see a
body in motion, we cannot determine by its behavior how
many forces have concurred to produce its motion.

59. Triangle of forces. — Since in Fig. 39 BE = AC, the three
forces which are represented in the parallelogram by the lines A B
AC, and A E, are also represented by AB, BE, and AE, three
sides of a triangle ABE.

Hence, if two forces are represented by two sides of a triangle, the
third side will represent their resultant.

60. Polygon of forces. — If any
number of forces applied at a point
are represented by all the sides but one ^■=
of a polygon, the remaining side will
represent their resultant. Thus the
forces AB, AD, and AE (Fig. 41),
are represented respectively by the
sides A' B', B'C (=AD), and C'F'
( = A E) of a polygon A' B' C F' ^^^- *^-

(Fig. 42), that is completed by the side A'F' (= AF), which rep-
resents the resultant of the three forces.

MOLAR DYNAMICS.

61. Parallelopiped of forces. — If three forces not in the same

^ plane are applied at a point, they

"":^:^^^^ will form three edges of a par al-

---,---^^ iQlgplpQ^^ ^^(^ ffifj^^ diagonal of

'\^ '\ this solid which is concurrent with

'\ \ these edges will represent the re-

D^\W>^ ^'-,:^^ sultant of these forces. It will be

A B readily seen that the resultant of

^^^•^^- the forces A B, AD, and AC

(Fig. 43), is represented by the diagonal AE.

62. Resolution of forces. — Assume that a ball has an
acceleration in a certain direction A C (Fig. 44), and that one
of the forces that produces this acceleration is represented in
intensity and direction by the line A B ; what must be the
intensity and direction of the other force ? Since AC is the
resultant of two forces acting at an angle to each other it is
the diagonal of a parallelo-
gram of which AB is one
of the sides. From C, draw
CD parallel and equal to

B A, and complete the paral- ^ * '^

lelogram by connecting the

points B and C, and A and D. Then, according to the
principle of composition of forces, A D represents the inten-
sity and direction of the force which, combined with the force
A B, would move the ball from A to C. The component A B
being given, no other single force than A D will satisfy the
question.

Had the question been. What forces can produce the motion
AC? an infinite number of answers might be given. In a
like manner, if the question were. What numbers added
together will produce 50 ? the answer might be 20 + 30,
40 + 10, 20 -p 20 -|- 10, and so on, ad infinitum ; but if the
question were, What number added to 30 will produce 50 ?
only one answer could be given.

RESOLUTION OF FORCES »

It is often necessary to resolve a force in order to ascertain the
effective force in a certain direction. Thus when boat sails are
exposed obliquely to the wind, the pressure effectual in moving the
boat is only a component of the whole force of the wind. The line
af (Fig. 45) represents the force of the wind acting on the sail c cZ at
the point a. Resolving this force we obtain the components 2

(normal to the sail) and! (a useless component called a tail wind).
The boat does not move in the direction of the pressure on its sail,
because it is more easily moved lengthwise than breadthwise. Hence
the normal pressure must be resolved into two components, one 4
along the direction of least resistance, i.e. the direction of easy
motion, the other 3 at right angles to it. The latter component
does tend to cause a slow broad-side motion called leeway, but this
may be partly counteracted by a deep keel or a center-board so that
the boat will sail approximately along the line a b.

Problems.

1. Draw upon paper pairs of lines making about the s^me angles with
each other as A B and A C in the four diagrams, Fig. 46, and having

B B

about the same directions ; assign arbitrarily numerical values to each
component, drawing to scale, and find the direction and the numerical
value of the resultant of each pair of components.

64 MOLAR DYNAMICS.

2. a. Find the intensity of the resultant of two forces acting at an
angle of 45°. 6. Find the intensity of their resultant when they act at an
angle of 150°. (The pupil will require either a pair of dividers or a pro-
tractor. He will do well to learn to use both in measuring angles.)

3. a. A heavy rock rests upon a smooth plane ; two men, A and B,
pull the rock by means of ropes attached to it, A with a force of 100 lbs. ,
B with a force of 150 lbs. If A pull toward the north and B toward the
south, what will be the resultant ? 6. If A pull toward the east and B
toward the south, what will be the resultant ? c. In the last case, if the
easterly acceleration at a certain instant is 10 feet per minute, what is
the southerly acceleration at the same instant ? d. In what direction
should they pull the rock to give it the maximum acceleration ? e. If A
pull it 25° S. of E., in what direction and with what force may B pull it
that the resultant may be directly east ? /. Give a different answer to
the last question.

4. On a scale of 1 cm = 1 K, represent a force of 5 K acting north-
ward on a point A.

5. On a scale of 1 cm = 1 K, represent forces of 4 K, 6 K, and 8 K,
acting simultaneously on point A in directions respectively as follows :
N., N. E., and S. E. Find their equilibrant.

6. A ship is sailing N. N. E. at the rate of 12 knots per hour. Find
its northerly and easterly velocities.

7. Find, both by construction (of parallelogram) and by calculation,
the intensity of two equal forces acting at right angles to each other, that
will support a weight of 15 pounds.

8. A sailor climbs a mast at a uniform rate of 5 feet a minute while
the vessel moves forward at the rate of 15 feet a minute ; what is his
actual velocity ?

9. On a scale of I of one inch = 10 lbs. , represent a force of 80 lbs.
Eesolve this force into two forces one of which shall act at an angle of
30° with the given force. Determine the numerical intensities of each of
the components.

10. Show by construction that a north-east wind is made up of a

north and an east wind, each — p of the actual velocity of the wind.

11. If two lines AB, C A represent two forces acting on point A, the
one toward and the other from it, show how to find the resultant.

12. Find the resultant of two equal forces of P lbs., the angle between
them being 120°.

13. Two rafters, making an angle of 60°, support a chandelier weighing
90 lbs. ; what is the pressure along each rafter ? Ans. 51.96 pounds.

LAWS OF MOTION. 65

Section VII.
DISCUSSION OF Newton's three laws of motion.

63. Laius of motion. — The science of dynamics rests on
certain fundamental principles termed the Laws of Motion,
first clearly stated by Newton in the ^' Principia " two cen-
turies ago, and verified by universal experience. The laws as
given in this text-book are as originally enunciated by New-
ton, with very slight verbal modifications in conformity to
modern terminology.

First Lata : A body at rest remains at rest, and a body in
motion continues to move with constant speed in a straight line,
unless acted upon by some external unbalanced force.

This law may be paraphrased as follows : A body under the
action of no force, or of balanced forces, is either at rest or in
uniform motion ; if it be at rest it will remain at rest, and if
it be in motion, its motion will be in none other than a straight
line, and its velocity will never change.

Motion unobstructed is perpetual. "Is perpetual motion
possible ? " has been often asked. The answer is simple, —
yes, more than possible, often necessary, if no force interfere to
prevent. (Example : the motions of the planets.) On our
earth we have no instances, for resistances such as friction,
resistance of the air, etc., are continually opposed to all
movements of terrestrial bodies. On this account we find
that force is required to perpetuate the motion of all bodies
with which we deal, and we fall readily into the fallacy that
force is necessary to Tnaintain motion, which the First Law
distinctly contradicts.

The clause " Unless acted upon by an external force " virtu-
ally states that "All matter is inert," i.e. that bodies of mat-
ter are utterly incapable of putting themselves in motion or
stopping themselves ; the inability is called inertia. Inertia

66 MOLAR DYNAMICS.

may be defined as that property of matter in virtue of which
external force is required to produce a change in momeyitum.
It is the sole unalterable property of matter.

The terms mass and inertia are often used interchangeably to
denote a quantity proportional to the unbalanced force required
to produce a given change in the velocity of a body in a given
time. It is known that all bodies unobstructed by the air fall
with the same velocity irrespective of their masses or inertia. But
to produce equal acceleration in equal times requires forces propor-
tional to the masses; it follov^s, then, that at the same locality
weight and mass {or inertia) are proportional. Hence we compare
masses by comparing their weights.

The somewhat vague yet common expressions "to overcome
inertia" and "to destroy inertia" mean to produce a certain
change of mass-motion {i.e. momentum), and may signify either an
increase or a decrease of the same.

Second Laiv : Change of momentum is in the direction in
which the unbalanced force acts, and is proportional to its inten-
sity and to the time during luhich it acts.

It will be seen that tliis law (except as regards direction)
is contained in the formula MV=Ft (p. 37) which has
already been developed. This formula virtually asserts that
where there is no force there is no change of momentum
(i.e. if F = 0, Ft = 0). Hence the First Law of Motion is a
deduction from the Second.

This law declares, by implication, (1) that an unbalanced
force in a given time alivays produces exactly the same change
of momentum regardless of the mass of the body ; that an
unbalanced force never fails to produce a change of momentum,
hence any force, however small, can move any body of however
great mass. For example, a child can move a body having a
mass equal to that of the earth, provided only that the motion
of this body is not hindered by a third body. Moreover, the
quantity of momentum that the child can generate in this
immense body in a given time is precisely the same as that

LAWS OF MOTION. 67

which he would generate by the exertion of the same force for
the same length of time on a body having a mass of (say) 10
pounds. Momentum is the product of mass into velocity;
so, of course, as the mass is large, the velocity acquired in a
given time will be correspondingly small. The instant the
child begins to act, the immense body begins to move. Its
velocity, infinitesimally small at the beginning, would increase
at an almost infinitesimally slow rate, so that it might be
years before its motion would become perceptible.

It is easy to see how persons may get the impression that
very large masses are immovable except by very great forces.
The erroneous idea is acquired that bodies of matter are capa-
ble of resisting the tendency of forces to cause motion, and
that the greater the mass, the greater the resistance (" quality
of not yielding to force," Webster). The fact is, that 7io body
of whatever mass can resist motion ; in other words, " a body
free to move cannot remain at rest under the slightest unbalanced
forceP But as tiine is always required to generate change of
momentum, there arises thence a deceptive appearance of
resistance or holding back.

This law declares by implication, (2) that a force acting
on a body in motion -produces just the same effect as if it were
acting on the same body at rest, for no reference is made in the
\2iW to the state of the body acted upon.

Experiment. — Draw back the rod d (Fig. 47) towards the left, and
place the detent-pin c in one of the slots. Place one of the brass balls on
the projecting rod, and in contact with the end of the instrument, as at A.
Place the other ball in the short tube B, Raise the apparatus to as great
an elevation as practicable, and place it in a perfectly horizontal position.
Eelease the detent, and the rod, propelled by the elastic force of the
spring within, will strike the ball B, projecting it in a horizontal direc-
tion. At the same instant that B leaves the tube and is free to fall,
the ball A is released from the rod, and begins to fall. The sounds
made on striking the floor reach the ears of the observer at the same
instant ; this shows that both balls reach the floor in sensibly the same

MOLAR DYNAMICS.

time, and that the horizontal motion whicli one of the balls has does
not affect the time of its fall, i.e. does not modify the effect of the force
of gravity.

The law implies, (3) that if two or more forces act on a
body, each ^produces its own change of monientuTn in its own
direction independently of the others. It declares, what we have
previously learned, that the operation of compounding forces

Flo. 47.

is just the same as that of compounding motions which the
several forces tend to produce in the same time, hence the
apparatus, Fig. 6, illustrates either the composition of motions
or the composition of the forces by which the motions are
produced.

Third Laiu: To every action there is an equal and opposite
reaction.

Previous to the announcement of the Laws of Motion our
studies have been such as to prepare us both to understand
and accept them. We have learned that there are always
two bodies or two parts of the same body oppositely affected
by every force. When the double aspect of a force, i.e. its

LAWS OF MOTION. 69

mutual action between two portions of matter, is considered,
it is customary to speak of the force as a stress. Illustrations
of stress are tension in a stretched rubber band and pressure
exerted between two bodies in contact when compressed. All
force is of the nature of a stress and the Third Law of Motion
virtually declares that evei^y action hetiueen tivo bodies is a
stress. When the effect of the action upon only one of the
two bodies is under consideration, the action is commonly
spoken of as a force.

It remains to show that action and reaction are equal. That
they are equal is deducible from the First Law, for if they
were unequal, then, when there is an action between two
parts of the same body, there would not be equilibrium. That
is, there would be an unbalanced force, which would cause
the body to move with accelerated velocity — a thing which
is explicitly contradicted by the First Law.

If action and reaction were not equal there might be a
possibility that a person might raise himself by pulling on
the soles of his feet or the hair of his head; that a vessel
might be propelled in a calm by blowing against its sail with
a powerful bellows (operated by steam) located on the deck
of the same vessel ; that a person sitting in a buggy might
give himself a ride by pressing his feet against the dasher ;
that a person might advance, i.e. move his center of mass,
without the earth beneath him ; that a bird might fly without
the external air to act on.

In case the two bodies are free from the action of resisting
forces the law implies that the momenta generated by the
action and reaction are equal.

The application of this law is not always obvious. Thus,
an apple falls to the ground in consequence of an action
between the apple and the earth. The motion of the earth
toward the apple is imperceptible. But this is because
the mass of the earth is enormously greater than that of

70 MOLAR DYNAMICS.

the apple, and its velocity, for an equal momentum, is
proportionately less.

Exercises.

1. a. Why does not a given force, acting the same length of time, give
a loaded car as great a velocity as an empty car ? b. After equal forces
have acted for the same length of time upon both cars, and have given
them unequal velocities, which will be the more difficult to stop ?

2. a. The planets move unceasingly ; is this evidence that there are
forces pushing or pulling them along ? b. None of their motions are in
straight lines ; are they acted upon by external forces ?

3. A certain body is in motion ; suppose that all hindrances to motion
and all external forces be withdrawn from it, how long will it move ?
Why? In what direction? Why? With what kind of motion, i.e.
accelerated, retarded, or uniform ? Why ?

4. Explain how rotating lawn-sprinklers are kept in motion.

5. When you leap from the earth, which receives the greater momen-
tum, your body or the earth ?

6. When you kick a door-rock, why does snow or mud on your shoes
fly off?

7. If a man in a boat move it by pulling oh a rope at one end, the
other end being fastened to a post, how is the boat put in motion ?
Would it move either faster or slower if the other end were fastened to
another boat free to move, the man exerting the same force ?

8. An ounce bullet leaves a gun of mass 8 pounds with a speed of 800
feet per second. What is the maximum speed of the gun's recoil ?

9. Suspend two balls of soft putty of equal mass, A and B (Fig. 48).
Draw A to one side, and let it fall so as to strike B. Both balls will then
move on together ; with what momentum compared with A's momentum
when it strikes B ?

10. What will be the momentum of each ball after A strikes B, com-
pared with A's momentum when it strikes B ?

11. How will their velocity compare with A's velocity when it strikes B?

12. Raise A and B equal distances in opposite directions, and let them
fall so as to collide. Both balls will instantly come to rest after collision.
Show that this result is consistent with the third law of motion.

13. Substitute for the inelastic putty balls, ivory billiard balls, which
are highly elastic. Let A strike B. Then B goes on with A's original
velocity, while A is brought to rest. Show that this result is consistent
with the third law of motion.

LAWS OF MOTION.

71'

14. Suspend four ivory balls, C, D, E, and F. Let C strike D. D
receives all of C's momentum, instantly communicates it to E, and E to
F. F, liaving nothing to which to communicate the momentum, moves
with C's original velocity. Trace the actions and reactions throughout.

15. What would happen if the four balls were inelastic ?

IG. A sliell at rest bursts into two parts, the smaller being one-third of
the whole ; what is the ratio of the initial velocities of the parts ?

17. a. Can any body, animate or inanimate, by any action confined to

A B

A' B' C'
Fig. 48.

6666

C D E F

itself, i.e. between component parts of itself, put itself in motion or stop
itself ? 6. How can a body put itself in motion ?

18. A child sits upon a sled. The sled is suddenly started and the
child is left sitting on the ice. a. Is this due to the inertia of the child ?
h. Is it due to a resistance which the child's body offers to a force tending
to put it in motion, or to the inadequacy of the force transmitted to it
through the sled to give its mass in the same time an equal velocity with
the sled ?

19. Why do not heavy bodies fall faster in a vacuum than light
ones ?

20. Take equal masses of wood and lead ; which weighs more ?

21. A stone falls from the top of a railway carriage which is moving at
the rate of one-half of a mile a minute. Disregarding the resistance of
the air, find what horizontal distance and what vertical distance the stone
will have passed through in one-tenth of a second. Ans. 4.4 ft. ; .16 ft.

"72 MOLAR DYNAMICS.

Section VIII.

APPLICATIONS OF THE LAWS OF MOTION. CURVILINEAR
MOTION.

64. How Gui'vilinear motion is produced. — Motion is curvi-
linear when its direction changes at every point. But according
to the first law of motion, every moving body proceeds in a
straight line unless compelled to depart from it by some
external force. Hence curvilinear motion can be produced
only by an external force acting continuously upon the body
at an angle to the straight path in which the body tends to
move, so as constantly to change its direction. In case the
body moves in a circle, this force acts at right angles to the
path of the body or towards the center of motion ; hence this
deflecting force has received the name of central force.

Thus, suppose a ball at A (Fig. 49), suspended by a string
from a point d, to be struck by a bat,
in a manner that would cause it to
move in the direction Ao. At the
same time it is restrained from taking
that path by the tension of the string,
which operates like a force drawing
it toward d. It therefore takes, in
obedience to the two forces, an inter-
mediate course. At c its motion is
in the direction C7i, in which path it
would move but for the string, in accordance with the first
law of motion. Here, again, it is compelled to take an inter-
mediate path. Thus, at every point, the tendency of the
moving body is to preserve the direction it has at that point,
and consequently to move in a straight line. The only reason
it does not so move is that it is at every point forced from its
natural path by the pull of the string. But if, when the ball

MAGNITUDE OF CENTRAL FORCE. 73

reaches the point i, the string be cut, the ball, having no
force operating to change its motion, continues in the direction
in which it is moving at that point, i.e. in the direction ih,
which is tangent to its former circular path.

65. Magnitude of central force for bodies moving in circular
jyaths.

Experiment 1. — Cause a ball to revolve around your hand by means of
a string attached to it and held in the hand. Observe closely every phase
of the operation. First you make a movement as if to project the ball
in a straight line. Immediately you begin to pull on the string to prevent
its going in a straight line. Under the continuous influence of these two
forces in a short time the ball acquires great speed. You may now cease
to exert any projecting force, and simply keep the hand still ; but as the
ball has acquired a motion, and all motion tends to be in a straight line,
you are still obliged to exert a pulling force to deflect it from its path.
Observe that, as the velocity of the ball is retarded by the resistance of
the air, the pulling or deflecting force which you are obliged to employ
rapidly diminishes.

To satisfy yourself that the ball tends to move in a straight line, let go
the string or cut it, and the ball immediately moves off in a straight line,
or simply perseveres in the direction it had at the instant the string was
cut. Observe that the ball appears while rotating to be pulling your hand ;
but you know that all the force concerned originates in yourself, and that
this apparent pull on the part of the ball is only the effect of the reaction
of the force which you exert on the ball. This reaction is erroneously
called "centrifugal force. "i

Every revolving body affords an example of central force
and centrifugal tendency. Hence we say that every revolving
body tends to fly away from the center (not radially, however,
but tangentially), and a central force is required to keep the
body in its circular path.

When you swing the ball about your hand you discover
that the force of the pull increases with the velocity, and
more rapidly than the velocity. Careful observations have

* There is no centrifugal force. The only force exerted is the central force, which
is of such a magnitude as to change the direction of the momentum just fast enough
to keep the body moving in a circle.

MOLAR DYNAMICS.

determined that for bodies revolving in circular orbits the
central force varies as the mass of the body, as the square of its
velocity, and inversely as its distance from the center.

Let a point move uniformly in the circular path P Q (Fig. 49a),
traversing the distance P in time t ;
then OP = v^l). If P be very near
O, the deflection T P from a straight
line due to the central force is ap-
proximately equal to ON. If a be
the acceleration towards the center
due to this force, ON = ia^2(2).
But by geometry 0P2z=0N-0D.
Comparing (1) and (2), we get v-t^

1)2

= |a^2.2r, ora = — . Since E ==
m a, we get F = m — .

The farther a point is from the axis of motion of a rigid
body, the farther it has to move during a rotation ; con-
sequently the greater its velocity. Hence, bodies situated at
the earth's equator have the greatest velocity, due to the
earth's rotation, and consequently the greatest tendency to fly
off from its surface. The effect of this is to neutralize, in
some measure, the force of gravity. It is calculated that a
body weighs about 2^9 less at the equator than at either pole,
in consequence of the greater centrifugal tendency at the
former place. But 289 is the square of 17; hence, if the
earth's velocity were increased seventeenfold, objects at the
equator would weigh nothing, i.e. the centrifugal tendency
would be equal to their weight.

The attraction between the sun and the earth causes these
bodies to move in curvilinear paths, performing what are
called annual revolutions. Were it not for this mutual
attraction (and the attraction of the other celestial bodies),
the motion of both these bodies would be eternally in straight

MAGNITUDE OF CENTKAL FORCE.

Fig. 50.

lines, but in consequence of their mutual attraction both
rotate about a point C (Fig. 50), which is the center of mass
of the two bodies considered as one body (as if connected
by a rigid rod).^ If
both bodies had equal
masses, the center of
gravity and center of
motion would be half-
way between the two
bodies ; but as the mass of the earth is less than that of the
sun, so its velocity and distance traversed are proportionally
greater. In reality the center of motion C is within the sun
near the edge toward the earth.

Experiment 2. — Apply the frame T (Fig. 51) to any rotating appa-
ratus as E (Fig. 52) so that it may be rotated about its axis d. The rod c
passes tlirough the balls a and 6 loosely so that the latter are free to
slide along the rod. The two balls are connected by a string so that
they are compelled to rotate as one body or one system of bodies.

Fig. 51.

The mass of a is twice that of h. Eotate the system, and show that there
is equilibrium in the system only when the center of h is twice as far from
the axis of rotation as the center of a. How does this verify the above
law ? While there is rotation, is there tension in the string connecting
the balls ? What is the cause of an action between the balls ? Ball a
pulls ball h ; what is the effect of this pull on 6 ? What is the effect of
the reaction on a ? Is there a similar action between the sun and earth in

1 Strictly speaking, the earth does not revolve around the sun any more than the
sun around the earth ; but both rotate about their commoh centroid.

MOLAR DYNAMICS.

their annual revolutions ? By what name is the action known ? If the
sun or the earth were instantaneously annihilated, state what would
happen to the other body if it were left entirely free, i.e. if its motion
were not affected by other bodies in the universe ?

Fig. 52.

Experiment 3. — Arrange some kind of rotating apparatus, e.g. E
(Fig. 52). Suspend a skein of thread a (Fig. 53) by a string, and cause

it to rotate ; it assumes the shape of
the oblate spheroid a\ Mount a glass
globe G (Fig. 52) about one-tenth
full of colored water, and rotate.

C,— «^w . . ». ^ The liquid gradually leaves the bot-

I '''^^s'^nS^ II /{l» n» ^^°^' ^^^®^' ^^^ forms an equatorial
d ^Pfc n B 11 wlill////Jr ring within the glass. This illustrates

the probable method by which the
earth, on the supposition that it was
once in a fluid state, assumed its
present spheroidal state. (Explain.)
Pass a string through the longest diameter of an onion c, and cause
it to rotate ; the onion gradually changes its position so as to rotate on
its shortest axis.

It can be demonstrated mathematically, as well as experi-
mentally, that a freely rotating body is in stable equilibrium

Fig. 53.

THE PENDULUM. 77

only when rotating about its shortest diameter; hence the
tendency of a rotating body to take this position.

Qyie&tions.

1. a. What is the cause of the stretching force exerted on the rubber
cord when you swing a return ball about your hand ? 6. Suppose that
you double the velocity of the ball ; how many times shall you increase
this stretching force ?

2. In what way can the tension in the string (Fig. 51) be so much
increased as to break it ?

3. Why do wheels and grindstones, when rapidly rotating, tend to
break, and the pieces to fly off ?

4. On what does the magnitude of the pull between a rotating body
and its center of motion depend ?

5. Oj. Explain the danger of a carriage being overturned in turning a
corner. 6. How many fold is the tendency to overturn increased by
doubling the velocity of the carriage ?

6. Account for the curvilinear orbits of the planets.

7. How are their motions in their orbits and around their axes main-
tained ?

8. In what way should the rails be laid so as to neutralize the centrif-
ugal tendency of a railroad train going around a curve ?

9. State and explain the posture of a bicycle rider in turning a curve.

10. In what way is the weight of terrestrial bodies nullified in some
degree by the earth's motion ?

11. A circus rider going around a ring inclines inward so that the line
of direction of his body falls without his base. How is he supported ?

Section IX.

THE PENDULUM.

Experiment 1. — From a bracket suspend by strings leaden balls, as in
Fig. 54. Draw B and C to one side, and to different hights, so that B
may swing through a short arc, and let both drop at the same instant.
C moves much faster than B, and completes a longer journey at each
swing, but both complete their swing, or vibration, in the same time.

Hence, (1) the time occupied by the vibration of a penduliiin
is independent of the length of the arc. Of only very small arcs

MOLAR DYNAMICS.

may this law be regarded as practically true. The pendulum
requires a somewhat longer time for a long arc of vibration
than for a short one, but the difference becomes perceptible
only when the difference between the arcs is great, and then
only after many vibrations.

Experiment 2. — Set all the balls swinging ; only B and C swing
together ; the shorter the pendulum, the
I faster it swings. Make B 1™ long, and
F i™ long. Watch in hand, count the
J) vibrations made by B, It completes just
G 60 vibrations in a minute ; in other words,
it " beats seconds." A pendulum, there-
fore, to beat seconds must be 1"^ long
(more accurately in the latitude of Boston
at sea-level, .9935™, or 39.117 in.). Count
the vibrations of E ; it makes 120 vibra-
tions in the same time that B makes 60
vibrations. Make G one-ninth the length
of B ; the former makes three vibrations
while the latter makes one, consequently
the time of vibration of the former is one-
third that of the latter.

•'a B C

Fig. 54.

Hence, (2) the time of one vibra-
tion of a pendulum varies as the
square root of its length.
The length Z of a simple pendulum to swing in a time t, or
the time of swing for a length I, can be found from the
formulae :

.9935 X f\ whence t = ^

'^fornneters;

or Z = 39.117 X tK whence t

39.117

for I inches.

The isochronism of the pendulum is utilized in the measure-
ment of time, i.e. in subdividing the solar day into hours,
minutes, and seconds. The office of the pendulum in

ACCELERATING FORCE OF GRAVITATION. 79

clocks is to regulate the rate of motion of the works. The
balance-wheel replaces the pendulum in watches and some
clocks.

66. Determination of the accelerating force of gravitation at
any locality. — The time of vibration is less at a place where
the force of gravitation is greater because the accelerating
force for the same mass is greater and hence the pendulum
will move faster.

Hence it is apparent that by determining the time of
vibration of a pendulum ^ of the same length at different
distances from the center of mass of the earth (e.g. at the
top and bottom of a mountain, or at sea-level at different
latitudes), the relative value of g at these places, i.e. the
acceleration produced by gravitation, may be ascertained.
We have already learned that the acceleration at the same
locality is the same for all bodies regardless of their mass.

By experiments too difficult for ordinary school work, it has
been ascertained that (3) the time of vibration of a pendulurn
varies inversely as the square root of the force of gravitation
(upon which the value of g depends).

To sum up the above three laws of the pendulum, we have
the formula^

■4

whence g=^ — ^>
9 ^

in which I = length of pendulum ; t = time of one vibration
in seconds.

1 The following measurement of g was made with great care occupying months
by Mendenhall, at Tokio, Japan, in the year 1880. The latitude of this place is
N. 35° 41'; value of g at sea-level 9.7984™; length of seconds-pendulum 994.59'°™.
On the summit of a neighboring mountain 12,441 feet above the level of the sea, he
found the time of vibration of the same pendulum to be 1.000336 seconds. From this
he computed the value of ^ = 9.7886". He also calculated the attraction of the
mountain to be .00021 the attraction of the earth, and that if the mountain were
annihilated, at that altitude g would be equal to 9.7865 m.

2 The student may find the development of this formula in Chapter VII of
Maxwell's " Matter and Motion."

80 MOLAR DYNAMICS.

At the poles of the earth the length of a seconds-pendulum
is 99.62«™ and ^ = 983.2«'^ per second. At the equator,
I = 99.10^"^ ; g = 978.1^™ per second (Kohlrausch).

67. Center of oscillation.

Experiment 3. — Connect six balls, at intervals of 15cm, y^y passing a
wire through them, after the manner of pendulum A (Fig. 54). This
forms a compound pendulum composed of six simple pendulums. Set
A and B vibrating ; A vibrates faster than B, although their lengths are
the same. Why is this ? If A v^ere actuated only by the ball/, it v^^ould
vibrate in unison w^ith B. If the ball a were free, it would move much
faster than / ; but, as they are constrained to move together, the tendency
of a is to quicken the motion of /, and the tendency of / is to check the
motion of a. But e is quickened less than /, and d less than e ; on the
other hand, h is checked by / less than a, and c less than 6. It is apparent
that there must be some point between a and / whose motion is neither
quickened nor checked by the combined action of the balls above and
below it, and where, if a single ball were placed, it would make the same
number of vibrations in a given time that the compound pendulum does.
Shorten pendulum B, and find the required point. This point is called
the center of oscillation.

Every compound pendulum is equivalent to a simple pendulum,
tvhose length is equal to the distance between the center of oscil-
lation ^ and the point of suspension of the compound pendulum.
Inasmuch as the distance between the point of suspension and
the center of oscillation determines the rate of vibration^
whenever the expression length of pendulum is used it must
be understood to mean this distance. Strictly speaking, a
simple pendulum is a heavy material point suspended by a
weightless thread. Of course such a pendulum cannot actually
exist ; but the leaden ball, suspended by a thread, is a near
approximation to it.

Experiment ^. — From the frame (Fig. 54) suspend at one of its ends
a lath (AB, Fig. 65) 1^ long. Find, as above, its center of oscillation.-
It will be found to be about two-thirds the length of the lath below
the point of suspension. Attach a weight to the lower end of AB ;

1 The center of oscillation may be defined as that point in a pendulum at which,
if its entire mass were collected, its time of vibration would be unchanged.

CENTER OF PERCUSSION.

its vibrations are now slower, and the simple pendulum B must be
lengthened to vibrate in the same time as the lath and weight ; hence the
center of oscillation of the lath is lowered by the addition of the
weight. Move the weight up the lath ; the vibrations are quick-
ened. (What is the office of a pendulum bob ?)

Experiment 5. — Remove the weight, bore a hole through the
lath at its center of oscillation C, and, passing a knitting-needle
through the hole, invert the lath and suspend it by the needle.
The pendulum is now apparently shortened, and we naturally
expect that its vibrations will be quicker than when suspended
from A. But the part B C now vibrates in opposition to the part
CA, rising as it sinks, and sinking as it rises. This tends to
check the rapidity of the vibrations of C A, and it is found that
the pendulum vibrates in the same time when suspended from ^^^- ^^•
C as when suspended from A. The point of suspension and the center
of oscillation are interchangeable.

68. Center of percussion.

Experiment 6. — Suspend the lath by a string attached to one of its

extremities, and with a club
strike it horizontally near its
upper extremity. This end of
the lath moves in the direction
of the stroke (A, Fig. 56), at
the same time causing a sudden
jerk on the string, which is felt
by the hand. Strike the lath
in the same direction, near its
lower extremity ; the upper
end of the lath now moves in
a direction opposite to the
stroke (B), at the same time
causing a similar jerk of the
string. Next strike the lath successively at points higher and higher
above its lower extremity ; it is found that the jerk on the string becomes
less till the center of oscillation is reached, when no pull on the string is
felt, and neither end of the lath tends to precede the other, but both
move on together (C). The full force of the blow is spent in moving the
stick, and none is expended in ]3ulling the string. This point is called
the center of percussion.

Fig. 56.

MOLAR DYNAMICS.

The center of percussion is coincide^it with the center of
oscillation. It is the point where a blow, given or received, is
most effective, and produces the least stress upon the support
or axis of motion. The base-ball player soon learns at what
point on his bat he can deal the most effective blow to the
ball, and at the same time feel the least tingle in his hands.

69. Demonstration of the earth'' s rotation on its axis.

ball and string pendulum be set in vibration in a certain plane,

then by virtue of the First Law of
Motion it will continue to vibrate
in the same plane even if the string
is twirled so that the ball rotates
on its axis. If the pendulum be
suspended from the ceiling of a
cabin in a vessel and the vessel be
turned completely around, the
plane of vibration will not be
changed. A hammock suspended
on deck of a ship retains the nor-
mal position independently of the
roll and rock of the ship, i.e. the
hammock does not swing, but the
ship supporting the hammock
swings. Now if a graduated cir-
cle be placed just beneath the ball,
as the vessel turns about the ball
will cross the circle at different
points and will appear to be chang-
ing its plane of vibration ; but it
is evident that this appearance is
deceptive, and that the graduated circle must turn around with the vessel,
and that the pendulum is merely pointing out the angular motion of the
vessel. In the same manner if a pendulum were suspended at one of the
poles of the earth, in 24 hours every meridian of the earth would be
brought beneath it, and although it would not meanwhile change its
plane of vibration, it would appear to move from east to west, or in
opposite direction to that of the earth, at the rate of 15° per hour. At
the equator there would be no change. Between the equator and the
pole the change per hour would vary from 0° to 15° according to the

Fig. 57.

earth's rotation on its axis. 83

latitude. With a heavy metal ball and a wire as small as will support
the ball and very long (Fig. 57), one may successfully repeat this cele-
brated experiment, by which Foucault demonstrated the motion of the
earth on its axis.

Questions and Prohlems.

1. What is the length of a pendulum that beats half-seconds? Quar-
ter-seconds? That makes one vibration in two seconds? That makes
two vibrations per minute ?

2. State the proportion that will give the number of vibrations per
minute made by a pendulum 40 cm long.

3. Where is the center of percussion in a hammer or axe ?

4. At what point (disregarding the length and weight of the striker's
arm) should a blow be dealt with a bat of uniform dimensions when held
in the hand at one extremity ?

5. What change in the location of the center of percussion is produced
by making one end of a bat heavier than the other ?

6. Which end of a bat, the heavier or the lighter, should be held in
the hands ? Why ?

7. One pendulum is 20 inches long, and vibrates four times as fast as
another. How long is the other ?

8. a. What effect on the rate of vibration of a pendulum has the weight
of its bob ? 6. What effect has the length of the arc ? c. What affects
the rate of vibration of a pendulum ?

9. How can you quicken the vibration of a pendulum threefold ?

10. A clock loses- time. a. What change in the pendulum ought to
be made ? h. How would you make the correction ?

11. Two pendulums are four and nine feet long respectively. While
the short one makes one vibration, how many will the long one
make?

12. What is the time of vibration of a pendulum (39.09 -f 4 =) 9.77 in.
long?

13. The number of vibrations made by a given pendulum in a given
time varies as the square root of the force of gravity. Force of gravity
at any place is expressed by the value of g {L e. by the acceleration which
it produces), a. If at a certain place a pendulum 39.09 in. long make
3600 vibrations in an hour, and the value of g be 32.16 ft., what is the
acceleration at a place where the same pendulum makes 3590 vibrations
in the same time ? b. Which of the two places is nearer the centroid of
the earth ?

84 MOLAR DYNAMICS.

14. Suggest some way by which the force of gravity at different
latitudes and altitudes may be determined.

15. A pebble is suspended by a thread 2 ft. long ; required the number
of vibrations it will make in a minute.

Section X.

WORK AND ENERGY.

70. Work. — Whenever a force causes a change of motion
or maintains motion against resistance it is said to do work.
A force may act for an indefinite time without doing work ;
for example, a person may support a stone for a time and
become weary from the continuous application of force to
prevent its falling, but he does no work because he effects no
change of motion or position. A force to do work 'must effect
a change of position. Force and &pace are essentials of work.
Force without motion is not work ; motion without force is
not work. The planets move, but do not work. Let the person
supporting the stone exert a little more force, — the stone
will rise and work will be done. An unbalanced force always
does work.

The body that moves another body is said to do work upon it ;
and the body moved is said to have taork done upon it.

When the heavy weight of a pile-driver is raised, work is
done upon it ; when it descends and drives the pile into the
earth, work is done upon the pile, and the pile in turn does
work upon the matter in its path.

71. Energy. — By the energy of a body is meant '' its
capacity for doing work" (Maxwell). It is measured by the
quantity of work which the body possessing it is capable of
doing.; hence the unit of work is also the unit of energy. The
act of doing work consists in a transfer of energy from the
body doing work to the body on which work is done, as when
the wind propels a vessel ; or it consists in a transformation

KINETIC AND POTENTIAL ENERGY. 85

of one kind of energy into another kind, as when the pile
driver strikes the pile and the pile is forced into the earth.
Here, a ^Dart of the energy in each act is transformed into
heat, which we shall learn, farther on, is molecular energy.
Work, therefore, may be defined as the act of transmitting or
transforming energy.

"We are acquainted with matter only as that which may have
energy communicated to it from other matter,, and which may in its
turn communicate energy to other matter. Energy, on the other
hand, we know only as that which in all natural phenomena is con-
tinually passing from one portion of matter to another. ' ' (Maxwe ll. )

72. Kinetic and ^potential energy. Experience teaches that
every moving body can impart motion, therefore it can do
work upon another ; hence every moving body possesses energy.
The energy which a body possesses in consequence of its
motion is called kinetic (motion) energy. It is a property of
a moving mass only. It is capacity for doing work possessed
by a mass in virtue of its motio7i.

When a body is projected upward its kinetic energy dimin-
ishes as it rises and finally becomes nil, but it is not lost, for
it is regained as the body falls. Its energy becomes, while
rising, stored up in virtue of its higher position. Energy in
store, i.e. not in an active state, is called potential energy. It
is the capacity for doing work possessed by a mass in virtue
of its p)osition being such that it is possible for it to move, and in
virtue of the existence of a stress tvhich tends to move it. Hence
it is convertible into kinetic energy without the agency of any
additional work except to remove obstacles to the conversion.
Potential or positional energy implies force, or a tendency to
motion, as truly as kinetic energy implies motion.

Illustrations of energy in the potential state :

(1) A stone lying on the ground is devoid of energy. Baise
it and place it on a shelf ; in so doing you perform work upon

86 MOLAR DYNAMICS.

it. As you look at it lying motionless upon the shelf, it
appem^s as devoid of energy as when lying on the earth.
Attach one end of a cord to it and pass it over a pulley and
wind a portion of the cord around the shaft connected with a
sewing-machine, lathe, or other convenient machine. Suddenly
withdraw the shelf from beneath the stone. The stone moves ;
it communicates motion to the machinery, and you may sew,
turn wood, etc., with the energy given to the machine by the
stone.

The work done on the stone or the energy transmitted to
the stone in raising it, was not lost ; it was recovered while
the stone was descending. There is a very important differ-
ence between the stone lying on the ground, and the stone
lying on the shelf : the former is powerless to do work ; the
latter can do work. Both are alike motionless, and you can
see no difference, except an advantage that the latter has over
the former in having a position such that it can move. What
gave it this advantage ? Work. A body, then, may possess
energy due m^erely to advantage of position, derived always
from work performed upon it. We see, then, that energy may
exist in either of two widely different states. It may exist
in bodies by virtue of their actual motion, or it may exist in
bodies by virtue of their having an opportunity to move, as in
the stone lying on the shelf.

Possibly some will object that the work done is performed
by gravity, and not by the stone's energy ; that if this force
should cease to exist, the stone would not move if the shelf
were removed, and consequently no work would be done. All
this is very true, and it is likewise true that when the stone
is on the ground the same force of gravity is acting, but can
do no work simply because the position of things is such that
the stone cannot move. The energy which the stone on the
shelf possesses is due to the fact that its position is such that
it can move, and that there is a stress between it and the earth

KINETIC AND POTENTIAL ENERGY. 87

which will cause it to move. Both advantage of position and
stress are necessary, but the former is attained only by work
performed. The force of gravity is employed to do work, as
when mills are driven, by falling water ; but the water must
first be raised from the ocean-bed to the hillside by the work
of the sun's heat. The elastic force of springs is employed
as a motive power : but this power is due to an advantage of
position which the molecules of the springs have first acquired
by work done upon them.

We are as much accustomed to store up energy for future
use as to store up provisions for the winter's consumption.
We store it when we wind up the spring or weight of a clock,
to be doled out gradually in the movements of the machinery.
We store it when we bend the bow, condense air, or raise
any body above the earth's surface.

(2) Matter may possess potential energy in virtue of
chemical separation and chemical affinity, and the potential
energy is a measure of the work done in effecting the sepa-
ration. For example : the entire value of coal consists in its
potential energy, which was stored up by the work performed
through the agency of the sun's energy in separating the
carbon of carbon dioxide from the oxygen. Gunpowder pos-
sesses, in a dormant state, energy sufficient to do a quantity
of work, e.g. in blasting, which would require many laborers
a long time to do.

A body possesses 'potential energy when, in virtue of work done
upon it, it occupies a position of oAimntage, or its constituent
particles occupy positions of advantage, so that the energy ex-
pended caM he at any time recovered hy the return of the body to
its original position, or by the return of its particles to their
original positions.

73. Relation of energy to force and matter. — Our discussions lead
us to conclude that energy is a condition of matter, due either to its
motion or to its relation with other matter, in virtue of which the matter

88 MOLAR DYNAMICS.

is capable of doing work. Force i may be regarded as ' ' the measure
of the tendency of energy" to transfer or "to transform itself."
(Tait.)

Energy is never found except in association with matter. Hence
matter may be defined as the veliicle or receptacle of energy. The
First Law of Motion affirms that matter is simply passive, inert.

74. Fractical imits of work and energy. — Inasmuch as a
body's capacity to do work is dependent wholly upon the work
which has been done upon it^ it is evident that both work and
energy may be measured by the same unit. The practical
unit adopted is the work done or energy irri'parted in raising one
•pound through a vertical hight of one foot. It is called a foot-
pound. The metric unit is the work done or energy imparted
in raising 1 K a vertical hight of 1 m, and is called a kilogram-
meter. The kilogrammeter is equivalent to 7,2331 ft. lbs.
Since the work done in raising 1 pound 1 foot high is 1 foot-
pound, the work of raising 1 pound 10 feet high is 10 foot-
pounds, which is the same as the work done in raising 10
pounds 1 foot high ; and the same, again, as raising 2 pounds
5 feet high.

There are many kinds of work besides that of raising
weights. But since, with the same resistance, the work of
producing motion in any other direction is just the same as
in a vertical direction, it is easy, in all cases in which the
resistance and space through which the resistance is overcome
are known, to find the equivalent in work done in raising a
weight vertically. By thus securing a common standard for
measurement of work, we are able to compare any species of
work with any other. For instance, let us compare the work
done in sawing through a stick of wood by a man whose saw

1 •■' By a convenient form of speech a given force is said to act upon a given body
and to impart to it a given acceleration. It must be constantly borne in mind, how-
ever, that a force is not a physical entity, and can never be measured until we
already know, absolutely or by comparison, the mass acted upon and the acceleration
imparted to it." (Danteli,.) " Force is a mere phantom suggestion of our muscular
sense." (Tait.) " Energy has its price, force has not."

ABSOLUTE UNITS OF WORK.

must move 10 m against an average resistance of 12 K, with
that done by a bullet in penetrating a plank to a depth of
2 cm against an average resistance of 200 K. Moving a saw
10 m against 12 K resistance is equivalent to raising 12 K
mass 10 m high, or doing 120 kgm of work ; a bullet moving
2 cm against 200 K resistance does as much work as is re-
quired to raise 200 K mass 2 cm high, or 200 X .02^4 kgm
of work. 120 -H 4 = 30 times as much work done by the
sawyer as by the bullet.

75. Absolute units of work} — If force be measured in dynes,

^bsoIute

Gravitation
Fig. 58.

and distance in centimeters, the work done is expressed in a
C.G.S. unit called an erg. An erg is the luork done or energy
imparted by a force of one dyne working through a distance of
one centimeter.

In purely scientific investigations absolute units are em-
ployed.

1 The pupil will, perhaps, be assisted by the accompanying diagram (Fig. 58) in
his first attempts to acctuire and classify the units of force, energy, and work in the
several systems.

90 MOLAR DYNAMICS.

The following equivalents will be useful :

Gravitation units absolute units

1 gram-centimeter =c= g ergs,

1 kilogram meter =c= 100^000 g ergs,

1 foot-pound =c=: 0.13825 X 10^ X ^ ergs.

76. Formulas for calculating luork or energy imparted. —
Force and space (ov distance), being essentials of work (p. 84),
are necessarily the quantities employed in calculating work.
A given force acting through a space of one foot does a certain
quantity of work ; it is evident that the same force acting
through a space of two feet would do twice as much work.
Hence the general formula

W=fi, (1)

in which / represents the force employed, s the space through
which the force acts, and W the work done.

In case a force encounters resistance, the magnitude of the
force necessary to produce motion varies with the resistance
(Third Law of Motion). Often the work done upon a body is
more conveniently determined by niultijolying the resistance hy
the space through luhich it is overcome, and our formula becomes
by substitution of r (resistance) for / (the force which over-
comes it)

rs=W: (2)

For example, a ball is shot vertically upward from a rifle in a
vacuum ; the work done upon the ball (by the explosive force
of the gunpowder) may be calculated by multiplying the
average force (difficult to ascertain) exerted upon it, by the
space through which the force acts (a little greater than the
length of the barrel); or by multiplying the resistance to
motion offered by gravity, i.e. its weight (easily ascertained),
by the distance the ball ascends.

CALCULATING WORK OR ENERGY IMPARTED. 91

Let us calculate the energy stored in a bow by an archer
whose hand, in bending the bow by pulling on the string,
moves 6 inches (^ foot) against an average resistance of
20 pounds. Here rs ^20 X i^= 10 foot-pounds of work done
upon the bow, or 10 foot-pounds of energy stored in the bow.

77. Formula for calculating kinetic energy. — Suppose a body
to have a mass m and a velocity v ; it can do a definite
quantity of work before it is thereby brought to rest. If it
be moving upward a mutual work between it and the earth is
performed in destroying each other's momenta. If its velocity
be such that it will rise to a hight s, then its kinetic energy
is such that it will do m g s absolute units of work, or

E]c (Kinetic energy) ^ m g s. (1)

We may find, then, to what vertical hight a body having
a given velocity would rise if directed upward, and from
formula (1) determine its kinetic energy ; but a formula may
be obtained which will give the same result with less trouble ;
thus, substituting g for a in the formula v = at (page 10), we
have v=-gt\ whence

t=-- , or t~ = —^-

Again 5 = -^^/?^^; substituting the value of t'^ in this equation
we have

g ^g

Substituting for s in equation (1) its value we have

^^^ = ^' (2)

a formula which will determine the kinetic energy of a body
in absolute units when its mass and velocity are known, since
the energy is the same whatever be the direction of the
motion.

92 MOLAR DYNAMICS.

Hence the kinetic energy of a body is half the product of its
mass by the square of its velocity.

If the result be desired in gravitation units, i.e. in gram-
centimeters or foot-pounds, the number of absolute units must
be divided by g^ since g ergs (980) are equivalent to one
gram-centimeter, or g foot-poundals (32.2) are equivalent to
one foot-pound.

Work done by a force is measured by the product of the numeric
of the force and that of the space s or [L] through which it acts.
Then since force [/] = [MLT— 2]^ the dimensional formula for
work [W] or energy [E] is, therefore, {fs=) [ML^T-^].

78. Energy co7itrasted luith momentum.'^ — It is evident from
formula (2) that when the mass (m) of a body remahis the
same, its energy is pvojiortional to the square of its velocity ;
while its momentum, as we have learned, is proportional to its
velocity. In other words, the effect of increasing the velocity
of a moving body would seem to be to increase its working
power much more rapidly than its momentum. Is this py^ac-
tically true ?

Experiment. — Fill a water-pail with moist clay. Let a leaden bullet
drop upon the clay from a hight of .5 m. Then drop the same bullet

1 Problem. — A bullet weighing 30 g is shot with a velocity of 98 m per second from

a gun weighing 4 K ; required the momentum and the energy of both the bullet and

the gun, and the velocity of the gnn. Solution: Using the kilogram, the meter, and

the second as units, the momentum of the ball is .03 x 98 = 2.94 units. If the ball

were shot vertically upward, its velocity would diminish 9.8 m per second ; so it would

98
rise ^ = 10 seconds, and, therefore, before its energy was expended, to a hight of

(§ 13) 4.9 m X 102 = 490 m. Hence, its energy at the outset is .03 x 490 = 14.7 kgm. By
the third law of motion the momentum of the gun must be just the same as that of
the ball, 2.94 units; its velocity is therefore 2.94^.4= .735m per second. Then

t = '-—— = .075 second ; the hight (supposing the gun to be raised vertically by the

impulse received) = 4.9 x .0752 = .02766 m ; and its energy = 4 x .02766 = .1102 kgm.

While, therefore, the momenta generated in the two bodies by the burning of the

14 7
powder are equal, the energy of the bullet is ' = 133^ times that of the gun.

(Why are the effects produced by the bullet more disastrous than those produced by
the recoil of the gun ?)

ENERGY CONTRASTED WITH MOMENTUM. 93

from a hight of 2 m, or four times the former bight, in order that it may
acquire twice the velocity. In the latter case it penetrates to four times
the depth that it did in the former.

So it appears that the energy of a moving body varies, not as
its velocity, but as the square of its velocity. Doubling the
velocity increases the energy fourfold, trebling the velocity
increases it ninefold, and so on ; but the corresponding
momentum is increased only twofold, threefold, etc. A bullet
moving with a velocity of 400 feet per second will penetrate,
not twice, but four times, as far into a plank as one having a
velocity of 200 feet per second. A railway train having a
velocity of 20 miles an hour will, if the steam be shut off,
continue to run four times as far as it would if its velocity
were 10 miles an hour. The reason is now apparent why
light substances, even so light as air, exhibit great energy
when their velocity is great.

Furthermore we have seen, p. 37, that momentum = 71^ F
=ft ; and again, p. 90, W (work done) or Ek (kinetic
energy imparted) ^= fs.

The momentum, then, imparted to a body is the product of
the force into the time it acts ; energy imparted is the product
of force into the space through which it acts. It is evident,
therefore, that force may be measured by the momentuim when
time is considered, and by the energy which it imparts when the
space is considered.

Questions and Problems.

1. Does the energy expended in raising the stones to their places in
the Egyptian pyramids still reside in the stones ?

2. What kind of energy is that contained in gunpowder ?

3. Can a person lift himself, or put himself in motion, without
exerting force upon some other body ?

4. a. Can a body do work upon itself ? b. Can a body generate
energy in itself, i. e. increase its own energy ?

94 MOLAK DYNAMICS.

5. a. Suppose that an average force of 25 pounds is exerted through
a space of 10 inches in bending a bow ; what amount of energy will it
give the bow ? b. What kind of energy will the bow, when bent,
possess ?

6. a. What amount of kinetic energy does a mass of 20 pounds moving
with a velocity of 300 feet per second, possess ? b. What amount of work
can the body do ?

7. How many fold is the kinetic energy of a body increased by
doubling its velocity ?

8. How high will twelve hundred foot-pounds of energy raise 100
pounds ?

9. A force of 500 pounds acts upon a body through a space of 20 feet.
One-fourth of the work is wasted in consequence of resistances. How
much available energy is imparted to the body ?

10. How much energy is stored in a body weighing 1,000 pounds, at a
bight of 200 feet above the earth ?

11. A horse draws a carriage on a level road at the uniform rate of
5 miles an hour. a. Does work accumulate ? b. What kind of energy
does the carriage possess ? c. Suppose that the carriage were drawn up
a hill ; would energy accumulate ? d. What kind of energy would it
possess when at rest on the top of the hill ? e. How would you calculate
the quantity of energy it possesses when at rest on top of the hill ?
/. Suppose that the carriage is in motion on top of the hill ; what two
formulas would you employ in calculating the total energy which it
possesses ?

12. How much work is done per hour if 80 K be raised 4 m per
minute ?

13. a. What energy must be imparted to a body weighing 50 g
that it may ascend 4 seconds ? b. How many times as much energy
must be imparted to the same body that it may ascend 5 seconds?
c. Why?

14. Compare the momenta, in the two cases given in the last question,
at the instants the body is thrown.

15. How much energy is stored in a body which weighs 50 K, at a
hight of 80 m above the earth's surface ?

16. How much kinetic energy would the same body have if it had a
velocity of 100 m per second ?

17. Suppose it to fall in a vacuum, how much kinetic energy would it
have at the end of the fourth second ?

18. If it should fall through the air, what would become of a part of
the energy ?

ACTIVITY. 95

19. A projectile of mass 25 K is thrown vertically upward with an
initial velocity of 29.4 m per second. How much energy has it ?

20. What becomes of its energy during its ascent ?

21. a. Compare the momentum of a mass 50 K having a velocity of
2 m per second, with the momentum of a body of a mass 50 g having
a velocity of 100 m per second, h. Compare their energies.

22. Which, momentum or energy, will enable one to determine the
amount of resistance that a moving body may overcome ?

23. Explain how a child who cannot lift 30 K can draw a carriage
weighing 150 K.

24. How many and what transformations take place during a single
swing of a pendulum ?

25. What quantity of energy will be expended if a force of 60 lbs.
move a body a distance of 20 ft. ?

26. A body of mass 30 lbs. moving with a velocity of 50 ft. per sec.
must do how much work before it stops ?

27. A United States 12-inch army gun, using a charge of 440 lbs. of
powder, throws a projectile of mass 1000 lbs. with an initial velocity of
1975 ft. per second, a. What quantity of energy is imparted to the
projectile ? h. The maximum range of this gun is 15 miles. If the
velocity of the projectile were uniform, in how many seconds would it
strike the ground ? c. If the projectile were directed vertically upward,
during how many seconds would it rise, the resistance of the air being
disregarded ? d. How high would it rise ?

28. a. A street car having a mass of 3 tons was moving at the rate of
6 miles per hour. The brakes were applied and stopped it in 4 seconds.
What was the average force exerted by the brakes ? 6. Suppose the car
to have been stopped in the space of 40 ft. , what was the average force
applied ?

79. Activity. — The activity (sometimes called the power) of
an agent is the rate at which it does or can do work ; or it is
the quantity of work it does in a unit of time, and is deter-
mined by the formula

. , ^. .^ - W (work)

In estimating the total quantity of work done, the time
consumed is not taken into consideration. The work done by

96 MOLAR DYNAMICS.

a hod-carrier in carrying 1000 bricks to the top of a building
is the same whether he does it in a day or a week. But in
estimating activity, as of a man, a horse, or a steam-engine,
in other words, the rate at which the agent is capable of doing
work, it is evident that time is an important element. The
work done by a horse in raising a barrel of flour 20 feet is
about 4000 foot-pounds ; but even a mouse could do the same
quantity of work in time.

The unit in which activity or rate of doing work is estimated
is called (inappropriately) a horse-power. A horse-power is
550 foot-pounds per second, 33,000 foot-pounds per minute,
or 1,980,000 foot-pounds per hour.

The logical unit of activity is a unit of work in a unit of
time, as one erg per second. The absolute unit of activity,
chiefly used in measuring electrical activity, is the loatt, or
ten million ergs per second. A moderate estimate of man's
activity is 100 watts.

1 erg per second -z- .0000001 watt,

1 horse-power =r= 746 watts,

1 foot-pound per minute -z- 226,043 ergs per second.

Activity being measured by the numeric of the work done per
unit of time, its dimensional equation is therefore [A] = [WT~i]
= [ML2T-3].

Questions and Problems.

1. For which is a truck- liorse valued, his energy or his activity ?

2. Do we speak of the activity or the energy of a steam-engine ?

3. Which do we apply to levers and machines in general, povv^er or
force ?

4. ' ' Energy is the power of doing work, ' ' Is this true ?

5. Shall we say that the activity, or the energy, of the horse is greater
than that of man ?

6. How much work can a 2 horse-power engine do in an hour ?

ACTIVITY. 97

7. a. What quantity of work is required to raise 50 tons of coal from
a mine 200 feet deep ? b. An engine of how many horse-power would be
required to do it in two hours ?

8. A car of mass 6000 k is drawn by a horse at a speed of 100 m per
minute. The index of the dynamometer to which the horse is attached
stands at 40 K. a. At what rate is the horse working ? b. Express the
rate in horse-power.

9. A dynamometer shows that a span of horses pull a plow with a
constant force of 70 K. What activity is required to work the plow if they
travel at the rate of 3 km per hour ?

10. What horse-power in an engine will raise 1,850,000 K 5m in an
hour ?

11. How long will it take a 3 horse-power engine to raise 10 tons
50 feet ?

12. How far will a 2 horse-power engine raise 1000 K in 10 seconds ?

13. A force of 10,000 dynes acting through a space of 100 meters per
second furnishes an activity of how many watts ?

14. The wind moves a vessel with a uniform velocity of 5 miles an
hour against a constant resistance of 2000 pounds. What activity is
furnished by the wind ?

15. If a 2 horse-power engine can just throw 1056 pounds of water to
the top of a steeple in two minutes, what is the hight of the steeple ?

16. A cannon ball of 10,000 g is discharged with a velocity of 45,000 cm
per second. Find its kinetic energy. Ans. 10125 X 10^ ergs.

17. In the last question, find the mean force exerted upon the ball by
the powder, the length of the barrel being 200 cm.

Ans. 50625 X 10^ dynes.

18. Supply the following ellipses by selecting appropriate words from
the following : viz. force, work, energy, activity. When — acts through
space — is performed, and — is imparted. The rate at which — is per-
formed determines the — of the agent. The — of a bullet flying through
vacant space. The — of a horse.- The ■ — of wind. The — of a bent
bow. What — must a bullet of mass 1 ounce have that it may rise 4
seconds ? What — is consumed by a steamer in crossing the ocean ?
What — is necessary that it may traverse 300 knots per day, and what
must be the average — exerted to overcome the resistances at the required
rate?

MOLAE DYNAMICS.

Section XI.

MACHINES.

80. Uses of maclmies.
Experiment 1. — Suspend,

as in Fig. 59, a fixed pulley, A, and a
movable pulley, B. Let the scale-pan
C counterbalance the pulley B, so that
there will be equilibrium . Suspend from
B two balls, LL, of equal weight, and
suspend on the side where the pan is,
a single ball, K, equal to one of the
former. The single ball supports the
two balls; i.e. by the use of the machine,
a force of 1 is enabled to balance a force
of 2. So far no work is done. Place a
very small weight in the pan ; the balls
LL rise, and work is done.

As the weight K plus a very small
weight causes the motion, we shall
regard this as the force (/); and as the
weights LL are the bodies moved (the
pulleys and pan being parts of the
machine may be disregarded), they may
be regarded as the resistance (r) over-
come, or the body on which work is
done. Measure the respective distances through which / acts and r moves
during the same time, r moves only one-half as great a distance as that
through which / acts; i.e. if r rise 2 feet, / must act through 4 feet.
Suppose that r is 2 pounds, then / is 1 + pounds. Now 2 (pounds) X 2
(feet) = 4 foot-pounds of work done on r. Again, 1 + (pounds) X 4 (feet)
= a little more than 4 foot-pounds of work (or energy) expended.

It thus seems that, although a machine will enable a small
force to balance a large force, when work is performed the
work applied to the machine is greater, rather than less, than
the work which the machine transmits to the resistance. The
work applied is greater than the work transmitted by the
amount of work wasted in consequence of friction and other

USE OF MACHINES. 99

resistances. So that hy the employment of a machine nothing
is gained in work, hnt something is aliuays lost.

What, then, is the advantage gained in using this machine?
Suppose that r is 400 pounds, and that the utmost force that
a man can exert is a little more than 200 pounds. Then
without the machine the services of two men would be required
to move the resistance ; whereas one man can move it with a
machine, but he will be obliged to move twice as far as the
resistance moves, a matter of little consequence in comparison
with the advantage of being able to do the work alone. The
advantage gained in this instance seems to be one of convenience.
Men, however, are accustomed to speak of it as "ct gain of
force ^^ (or more commonly and inaccurately, ^^of poiuer^^),
inasmuch as a small force overcomes a large resistance.

Experiment 2. — If instead of applying the small additional weight to
the pan, it he suspended from one of the balls LL, the weight of these
balls, together with the additional weight, becomes the cause of motion,
and K is the resistance. In this case there is a loss of force, because the
force employed is greater than the force overcome. Measure the dis-
tances traversed respectively by K and LL in the same time. K moves
twice as far as LL, and of course with twice the speed. There is a gain
of speed at the expense oi force.

It thus appears that, if it should be desirable to move a
resistance with greater speed than it is possible or con-
venient for the force to act, it may be accomplished through
the mediation of a machine, by applying to it a force propor-
tionately greater than the resistance. This apparatus is one
of many conti'ivances called machines, through the mediation of
which force can he applied to resistance more advantageously
than when it is applied directly to the resistance.

At present we deal with machines employed as means for
transmitting and modifying motion and force. Later we shall
consider machines whose function is to transform energy, such
as the steam engine, dynamo, etc.

100

MOLAR DYNAMICS.

Some of the many advantages derived from the use of
machines are :

(1) They may enable us to exchange intensity of force for
speed, or s])eed for intensity of force. A gain of intensity of
force or a gain of speed is called a mechanical advantage.

(2) They may enable us to emi^loy a force in a direction
that is more convenient than the direction in which the resis-
tance is to be moved.

(3) They may enable us to employ other forces
than our own muscular force in doing work;
e.g. the muscular force of animals ; the forces
of wind, water, steam, etc.

How are the last two uses illustrated in
Fig. 60? The pulleys employed are called
fixed pulleys, i.e. they have no motion except
that of rotation. Is any mechanical advantage
gained by fixed pulleys ? What is the use of a
fixed pulley ? Pulley B (Fig. 59) is a movable
pulley. What advantage is gained by means of
a movable pulley ?

81. General law of machines. —
From the experiments and discussion
above we derive the following for-
mula for machines :

fs = rs' -\r w, (1)

in which / represents the force applied, and s the distance
through which / acts ; r represents the resistance overcome,
and s' the distance through which its point of application is
moved ; lu represents the wasted work. A machine in which
there is no wasted work is a perfect machine. Such a machine
is purely ideal, as none exists. If in our calculations we
regard a machine as perfect (though subsequently suitable
allowance must be made for the wasted work), then our
formula becomes fs = rs', (2)

Fig. 60.

GENERAL LAW OF MACHINES. 101

whence r :/:*: s : s'; i.e. the force and resistance vary inversely
as the distances which their respective points of application move.
In other words, the ratio of the resistance to the force is the
reciprocal of the ratio of the distances which these points
move ; thus, if

r : f:= 4, then s' : s = J.

This law applies to machines of every description; hence it is
called the General or Universal Law of Machines. When r
is greater than /, there is a gain of intensity of force, and

- = the ratio of gain of intensity of force. When s' is greater

s'
than s, there is a gain of speed, and - =: the ratio of gain of

speed.

Since fs^ the work done upon a machine, is always greater
than rs', the work transmitted by the machine, we infer that
no machine creates or increases energy. No machine transmits
more energy than it receives. A machine may enable us to
gain intensity of force, but not energy. By taking s great
enough,. / can be made as small as we please ; in this case in
proportion as force is gained, time, distance, or speed is lost.

Formula (1) exhibits the relation between the work performed
upon the machine and the entire work transmitted and transformed
by the machine. But at any given instant while the machine is
in operation, it is evident that (see p. 91)

/s = i mv'^ + rs' + w. (3)

In formula (3) \mv'^ represents the kinetic energy of the moving
parts of the machine.

82. Efficiency of machines. — The efficiency of a machine is
a fraction, usually a per cent, expressing the ratio of the
energy given out by the machine and utilized to the total
energy expended upon the machine. The limit of the efficiency
of a machine is unity, which is the efficiency of an " ideal,"

102 MOLAR DYNAMICS.

or perfect, machine, in which no energy is lost. The object
of improvements in machines is to bring their efficiency as
near to unity as possible. For instance, if 50 foot-pounds of
energy be expended on a machine, and friction convert 8 foot-
pounds into heat and 5 foot-pounds be lost in consequence of
only a component of the working force being utilized, so that
the machine is able to give out only 37 foot-pounds, its
efficiency is f ^ = 74 per cent. If the friction can be reduced
one half and an improvement can be made in the machine
which will render the entire working force effective, then
there will be wasted only 4 foot-pounds of energy, and its
efficiency will be raised to f ^ = 92 per cent., and the quantity
of work which the machine will accomplish will be increased
in the ratio of 92 : 74.

83. Mechanical j^oivers. — Machines, however complicated
or complex, are largely composed of a few simple machines
long known as the ''mechanical poiuersJ' As usually given
they are the Lever, Wheel and Axle, Inclined Plane, Wedge,
Screw, Pulley, and Knee.

84. Experiments luith the lever.

Experiment 3. — Support a lever, as in Fig. 61, so that there shall be
unequal arms. Move W until the lever is balanced in a horizontal posi-
tion. Suspend (say) seven balls from the short arm (say) one space

from the fulcrum. Then from

w
.^^ i _^,___^_^_ ^_^oy . ^ i^ ^ '^ \-^_j ^j^ the other arm suspend a single

' (|) ball from such a place (in this

ife case seven equal spaces from the

) Q fulcrum) that it will balance the

Q seven balls. There is now equili-

FiG. 61. ' ' ' brium between the two forces.

Suppose the smaller force to be

increased a little and to produce motion; what mechanical advantage

{i.e. intensity of force or speed) would be gained by the use of the

machine ? What is the ratio of gain, the small additional force being

neglected ? How does this ratio compare with the ratio between the

length of the two arms ? For convenience we call the distance of the

EXPERIMENTS WITH THE LEVER.

103

KD=S

point of application of tlie force from the fulcrum the force-arm^ and the
distance of the resistance from the fulcrum the resistance-arm.

Suppose the small additional force to he applied to the short arm ;
what mechanical advantage would be gained ? What would be the
ratio of gain ?

While the general law of machines (p. 100) is always
applicable, its application is not always convenient, since, for
example, it necessitates putting the machine in motion in
order to measure s and s'
(the distances traversed
respectively by the points
of application of the force
and resistance in the same
time), an operation which
would be very difficult
and tedious in many
cases. Hence, a sidecial
law, one in which the
relation between the ratio
of gain and the ratio be-
tween certain dimensions
of the machine is stated,
is often more convenient
in practice. For example,
in our experiment with
the lever we discover that
E, : F : : force-arm: resistance-arm, i.e. the force and resistance
vary inversely as the lengths of their respective arms. Compare
this special law with the general law. Place the fulcrum at
other points in the lever, and thereby vary the length of the
arms, and verify by numerous experiments the special law
of levers.

Experiment 4. — By means of a pulley, D, so arrange (Fig. 62) that
both F and R may be on the same side of the fulcrum. First, place in

^W^£^

Fig. 62.

104

MOLAR DYNAMICS.

the pan weights sufficient to produce equilibrium in the machine (for
example, in this case, one ball). Then suspend weights at some point, as
A, and place other weights in the pan to coiinterbalance these. Verify
the law of levers. If A be the resistance, what mechanical advantage is
gained ? What is the ratio of gain ? If B be the resistance, what
mechanical advantage is gained ?

85. Wheel and axle. — The wheel and axle consists of two
cylinders having a common axis, the larger of which is called

Fig. 63.

Fig. 64.

the wheel, and the smaller the axle, as A and C (Fig. 63).
The wheel may be moved by the hand or by a string with a
weight attached to it.

The wheel is often replaced by a crank, as in the windlass,
or by a spoke, as in the capstan, and is thus employed in
hoisting apparatus, such as cranes, derricks, etc.

The wheel and axle viewed in section will be seen (Fig. 64)
to be only a modification of the lever which, unlike the latter,
may be continuous in its operation. C is the fulcrum, the
radius CA is the force-arm, and the radius CB the resistance-

INCLINED PLANE. 105

arm. The laws pertaining to this machine are virtually the
same as those of the lever. For example, when the force / is
applied to the wheel and the resistance r is at the axle,

R (radius of wheel) K (radius of axle)
1 1

C (circumference of wheel) C (circumference of axle).

86. Inclined plane. — Any plane surface not horizontal or
vertical, known as an inclined plane, may be used as a simple
machine for gaining intensity of force ; e.g. a plank resting
with one end on a cart body and the other on the ground, a
hill-side, or a road-grade. The gradient is the quantity of rise
per horizontal foot, or it is the ratio of the vertical rise to the
horizontal distance.

When a body is pressed against a hard, smooth surface, the
resistance offered by the surface is at right angles to the
surface. A body, e.g. a sphere, may be supported on a
horizontal surface, for the weight acting downward is counter-
acted by the upward reaction of the plane. But since on an
inclined plane the reaction is not vertically upward, a body
cannot rest on it without the aid of another force.

The mechanical advantage of this machine depends on the
principle of the resolution of a force into its components. Let

AB (Fig. ^^) be an inclined plane whose gradient is — ;^- Let

a be the centroid of the weight W (technically called the load).
The line of direction of the load is along the vertical ao, but
the pressure exerted upon the plane is in the direction ac, and
the reaction of the plane is in the direction ca. We may take
any length along the vertical as a 5 to represent the load W.
Draw he parallel to AB to meet ac. Complete the paral-
lelogram adhc with a^ as its diagonal. The force ah \&
thereby resolved into two forces, ac representing the pressure

106

MOLAE DYNAMICS.

upon the plane, and ad representing an unbalanced force
tending to move W along the plane B A. It is evident that
to produce equilibrium, i.e. to support the body on the plane
AB, a force equal to ad, but opposite in direction, must be

Fig. 65.

employed. Now it may be proved geometrically that the
triangles adh and BCA are similar; i.e.

da: ah : : CB : BA.

But da represents the force /necessary to produce equilibrium,
while 0^^ represents the load or resistance ?-; CB represents
the hight h, and BA the length I, of the inclined plane.
Therefore

f-.T-.-.h: I ovf=-r.

Hence a given force acting parallel ivith the direction of
inclination of an inclined plane, will support a weight as many
times greater than itself as the length of the inclined plane is
greater than its vertical hight. Corollary : with a given length
of inclined plane the greater its vertical hight, i.e. the steeper
it is, the greater /must be.

ANGLE OF REPOSE.

107

Suppose that a (Fig. ^^), the point of application of r, be
moved to V, how high is the load raised? Through what
distance does / act ? Show that the general law of machines
is applicable to this case.

87. Angle of repose.

When work is done, i.e. when the load is moved up the inclined
plane, a force greater than / must be employed, partly to overcome
friction, and partly to produce acceleration. On the other hand, if
a body be allowed to roll or slide down an inclined plane, the force
of gravitation overcomes friction and produces acceleration. Since /
diminishes with the angle of inclination of the plane, there must be
an angle at which / will be just equal to the friction. This angle is
called the angle of repose. If the angle of inclination be greater
than the angle of repose, the body is acted on by an unbalanced
force whose magnitude is determined by the inclination of the plane,
and the acceleration produced thereby will be proportional to the
unbalanced force. This suggests a way of " diluting " the action of
gravity, so as to be able to study the effects of a constant unbalanced
force, and a marble rolling down a smooth plane very slightly
inclined furnishes a very simple substitute for the falling weights
in Atwood's Machine.

88. TJie wedge is a triangular prism, used commonly as a
movable inclined pl^ne for moving great resist- .,
ances through short distances. It usually
consists of two inclined planes as AcP and ^^
BcP (Fig. QQ). The force applied acts in the
direction po and the resistance acts at right
angles to the planes, or in the directions Dm
and En. The force applied is of the nature
of a percussion, as that of a sledge ; besides,
friction and other resistances form so con-
siderable a factor in its use that no definite
law of any practical value can be given,
further than that, with a given thickness, fig. m.
the longer the wedge the greater the gain in intensity of
force.

108

MOLAR DYNAMICS.

89. The screw is another variety of the inclined plane, as
may be shown by winding a triangular piece of paper around
a cylinder, e.g. a lead pencil (Fig. 67). The hypotenuse will

Fig. 67.

Fig. 68.

form a spiral about the cylinder resembling the threads of
a screw.

In actual practice the screw consists oT two parts : (1) a
convex grooved cylinder, or scretv, S (Fig. 6S), which turns
within (2) a hollow cylinder, or 7iict, N. The concave surface
of the latter is cut with a thread corresponding
to the thread of the screw. The force is em-
ployed either to turn the screw within an immov-
able nut, or to turn the nut about a fixed screw.
In either case the force is usually applied to a
lever or wheel fitted either to the screw or to
the nut.

During a single rotation of the screw or nut,
the load or resistance is moved a distance equal
to the vertical distance between the correspond-
ing surfaces of two successive threads, usually termed the

Fig. 69.

THE KNEE OK TOGGLE-JOINT.

109

pitch of the screw, as ah (Fig. 69). Then in conformity to
the universal law of machines the force is to the resista7ice as
the distance hettveen the corresponding surfaces of two successive
threads is to the circumference of the circle described hy the
force.

Since a screw turning in a nut advances only its pitch distance at
each revolution, a finely cut ^-^

screw furnishes an instru- q f QS\ ^

ment well adapted to meas-
ure very minute distances.
For example, if the screw C
of the micrometer caliper
(Fig. 70) have a pitch of one
millimeter, and the thimble
D be divided near its end A
into one hundred parts so as
to register hundredths of a

revolution, it is evident that any object {e.g. a hair) placed in its
jaws at B can be measured to the hundredth of a millimeter.

90. The Jaiee or toggle-joi7it. — This machine also is em-
ployed where great pressure has
to be exerted through a small
space, as in punching and shear-
ing iron. Force applied at C
(Fig. 71) in the direction CD
will cause the jointed bars to be
brought into line with each other
and tend to push the objects at
the other extremities of the bars
apart. The nearer these bars
approach to a straight line, the
greater pressure will a given
Fig. 71. force produce.

110

MOLAR DYNAMICS.

Exercises.

1. a. When is a machine said to gain intensity of force? 6. When
is it said to gain speed ?

2. a. How is intensity of force gained by the use of a machine?
h. How is speed gained by the use of a machine ?

3. a. What is mechanical advantage? h. Give a rule by which the

Fig. 72.

mechanical advantage that may be gained by any machine may be
calculated.

4. Energy is applied to a machine at the rate of 250 ft. lbs. per minute
and it transmits 200 ft. lbs. per minute. What is its efficiency ?

5. Fig. 72 represents a pile-driver, a. How can the energy or the
work which the weight W can do when it is raised a given distance be
computed ? h. What benefit is derived from the use of the machine in

EXERCISES.

Ill

raising the weiglit ? c. Suggest some simple attachment to the machine
which would enable one man to raise the weight, d. Suggest some
attachment by means of which a horse could be employed to do the
work. e. What difference will it make whether the weight is raised

5 feet or 10 feet ? /. Illustrate, by means of this machine, what you
understand by force and energy.

6. a. What advantage is gained by a nut-cracker (Fig. 73) ? b. What
is the ratio of gain ?

7. a. What advantage is gained by cutting far back on the blades of
shears near the fulcrum (Fig. 74) ? "*Why ? 6. Should shears for cutting

Fig. 74.

metals be made with short handles and long blades, or the reverse ?
c. What is the advantage of long blades ?

8. The arm is raised by the contraction (shortening by muscular
force) of the muscle A (Fig. 75),
which is attached at one extremity
to the shoulder and at the other ex-
tremity B to the fore-arm, near the
elbow, a. When the arm is used, as
represented in the figure, to raise a
weight, what kind of machine is it ?
6. What mechanical advantage is
gained by it ? c. How can the ratio
of gain be computed ? d. For which
purpose is the arm adapted, to gain
intensity of force or speed ?

9. Is work done when the moment of the force applied to a lever is
equal to the moment of the resistance ? Why ?

Fig. 75.

112

MOLAR DYNAMICS.

Fig. 76.

A 10. If r (Fig. 76) be the fulcrum of

the lever C A, and A B represent, on a

■ scale of 1 cm = 1 k, a force applied at A,

what force applied at C in the direction

C D will produce equilibrium ?

11. With a wheel and axle a force
of 8 lbs. sustains a weight of 56 lbs. ; what is the ratio between the radii
of the wheel and of its axle ?

12. A capstan turned by two horses is used to draw a boat ; the horses
are attached to the levers 12 feet

from the axis of the capstan ; the
radius of the axle is 18 in. When
each horse pulls with a force of
1,000 lbs. what force is exerted
upon the boat ?

13. Suppose the screw in the
letter-press (Fig. 77) to advance
inch at each revolution, and a force
of 25 pounds to be applied to the cir-
cumference of the wheel &, whose
diameter is 14 inches. What pres-
sure would be exerted on articles
placed beneath the screw ?

14. A lever is 75 cm long ; where must the fulcrum be placed in order
that a force of 2 k at one end may balance 4 k at the other end ? What
will be the pressure on the prop ?

Fig. 77.

Fig. 78.

15. Two weights, of 5 k and 20 k, are suspended from the ends of a
lever 70 cm long. Where must the fulcrum be placed that they may
balance ?

EXERCISES.

113

16. If P (Fig. 78), weighing lib., is suspended 15 spaces from the
fulcrum of the steelyard, what weight (W) suspended 3 similar spaces the
other side of the fulcrum will balance it ?

17. How would you weigh out 6 pounds of tea with the same steel-
yard ?

18. a. A skid 12 feet long rests with one end on a cart at a hight of
3 feet from the ground. What force will roll a barrel of iiour weighing
200 lbs. over the skid into the cart ? h. What amount of work will be
required ?

19. a. Draw a line to represent an inclined plane. Find what is the
least force that will prevent a ball weighing 96 lbs. from rolling down the
plane. 6. Find the pressure which the ball will exert upon the plane.

20. An iron safe on trucks, weighing two tons, is prevented from
rolling down an inclined plane by a force of 250 lbs. What is the ratio
of the length of the plane to its hight ?

21. The gradient of an inclined plane is 1 ft. in 4 ft. To produce
equilibrium on this plane what relation must the force applied parallel
with the plane bear to the load ?

22. If the circumference of
an axle (Fig. 79) be 60 cm, and
the force applied to the crank
travel 240 cm during each revo-
lution, what force will be neces-
sary to raise a bucket of coal
weighing 40 k ?

23. Through how many me-
ters must the force act to raise
the bucket from a cavity 10 m
deep?

24. The truck (Fig. 80) is a lever ; the fulcrum is at the axle M of the
wheels. AB represents the line of direction of the load, i.e. the direction
in which the resistance acts ; and C D represents the direction in which
a force acts to produce equilibrium in the load in its present position.
a. What represents the force-arm ? b. What represents the resistance-
arm ? c. The force required to support the load is what part of the
load ? d. Would greater, or less, force be required if it were applied at
E instead of C ? Why ? e. How may the load be supported without any
force applied to the lever, the legs not touching the ground ? /. Would
its equilibrium in this position be stable or unstable ? Why ? g. Suppose
the feet F to rest upon the ground, how would the pressure of the load be
distributed between the feet and wheels ? h. Which is better suited for

Fig. 79.

114

MOLAR DYNAMICS.

moving heavy burdens, a wheelbarrow or a truck ? Why ? i. Suppose
that C D represents the supporting force and C G the force employed in
moving the load, how would the intensity and direction of the single
force that accomplishes both results be found ?

25. A plank 12 feet long and weighing 24 pounds is supported by two

Fig. 80.

props, one 3 feet from one end, and the other 1 foot from the other end.
What is the pressure on each prop ?

26. What must be the diameter of a wheel in order that a force of 20

Fig. 81.

Fig. 82.

pounds applied at its circumference may be in equilibrium with a resist-
ance of 600 pounds applied to its axle, which is 3 inches in diameter ?

27. How would you calculate the mechanical advantage gained by a
machine like that of Fig. 81 ? (On the axle A is an endless screw a, by
means of which motion is communicated from the axle to the wheel W.)

COMBINATION OF MACHINES.

115

28. a. Where is the fulcrum in a claw-hammer (Fig. 82) ? b. What
is the ratio of the mechanical advantage gained by means of it ?

29. In its technical meaning, a "perpetual motion machine " is not a
machine that will run indefinitely, but a machine which can do work
indefinitely without the expenditure of energy. Show that such a machine
is impossible.

Section XII.

COMBINATION OF MACHINES.

91. Comhination of pulleys. — As has been shown (p. 98),
mechanical advantage is gained with the movable pulley. In
Fig. 83, M, the cord passing around the movable pulley may
be supposed to be divided into two parts each supporting half

Fig. 83.

Fig. 84.

the load; the tension in each part is half the load, and the
supporting force P is half the load. In Fig. 83, N, the movable
block A contains two pulleys, the cord supporting the load is
divided into four parts, and P is one fourth of L ; and generally
with a combination of pulleys having a continuous cord

i.e. the force is equal to the load divided by the number of parts
of the cord supporting the movable block.

116

MOLAR DYNAMICS.

92. Combination of levers. — In the arrangement shown in
Fig. 84 there are three simple levers combined so as to form
one coni])oiind lever. The supporting force is applied at A :
the resistance applied to this simple lever at B is identical
with the force applied at A', and so on. Now

Continuous product of resistance-arms
f — I \^ ±- ii .

Continuous product of force-arms

A combination of levers similar to this may be seen in
scales used for weighing very heavy bodies such as the
so-called platform "hay scales," in which a, comparatively
small weight counterbalances the heavy load.

93. Combinations of the
ivheel and axle. — Fig. 85
represents a train of wheels
in gear. A train of wheels
being analogous to a com-
pound lever, the mechanical
advantage gained is obvi-
ously the ratio of the con-
tinued product of the radii
of the wheels to the con-
tinued product of the radii
Fig. 85. of the axles.

JE'xercises.

1. When there are four movable pulleys in one block, what length of
cord passes through the hands in raising a weight 6 inches ?

2. What force must a man weighing 180 pounds use, to support him-
self by means of the pulleys in Eig. 83, N ?

3. The circumference of the circle described by the end of the lever of
a screw is 9 feet, and there are three threads to the inch ; what pressure
will a force of 100 pounds exert ?

4. Suppose the lengths of the arms of the several levers in Fig. 84
bear the following relations to each other: 5:1,4:1, and 3:1; what force
applied at A will support 180 pounds at W ?

COMBINATION OF MACHINES.

117

5. a. Of how many and what simple machines does the crane (Fig. 86)
consist ? 6, How would you calculate the mechanical advantage gained
by it?

6. a. In what sense are machines ' ' labor-saving ' ' ?
is no machine labor-saving ?

6. In what sense

118 MOLAR DYNAMICS.

CHAPTER 11.

GRAVITATION, OR UNIVERSAL ATTRACTION BETWEEN

MASSES.

94. Gravitation is universal. — An unsupported body falls
to tlie earth. This is evidence of an action or stress between
the earth and the body. It has been ascertained by careful
observation that when a ball is suspended by a long string by
the side of a mountain, the string is deflected from the vertical
toward the mountain in consequence of an attraction between
the mountain and the ball. Delicate experiments show that
very small bodies tend to approach one another, and that the
tendency of a body to fall to the earth is but a single instance
of a tendency existing in all kinds and quantities of matter.

That there is an attraction between the sun and the earth,
and the earth and the moon, is shown by their curvilinear
motions. Tides and tidal currents on the earth are due to the
attraction of the sun and the moon. This attraction which
exists between all masses is called gravitation. When bodies
under its influence tend to approach one another, they are said
to gravitate. Since this attraction ever exists between all
bodies, at all distances, it is called universal gravitation. The
theory of universal gravitation was established by Newton.
No concept in science rests on a surer foundation.

95. Weight. — The attraction between the earth and a ter-
restrial body is called the weight of that body. The weight
of a body, therefore, is the measure of the force of attraction
between the body and the earth.

We do not know the cause of gravitation. Whether the
seat of the attraction or energy is, as the language in common
use indicates, in the bodies themselves, or whether it exists in

LAW OF GRAVITATION. 119

some medium which we may suppose to surround all bodies
and fill all intervening space, we do not know.^

96. Law of gravitation. — Methods first given by Newton
in the Princijna, but too elaborate for our purpose, have
established the fact that the magnitude of the attraction
between any two bodies depends upon two things, their
masses and the distance hetiveen their centroids. The Law of
Universal Gravitation is as follows : —

The attraction between every two bodies of matter in the
universe vaines directly as the 'product of their masses, and
inversely as the square of the distance between their centroids?

Eepresenting the masses of two bodies by m and m\ the
distance between their centers of mass by d, and the attrac-
tion by g, this relation is expressed mathematically thus :

g oc (varies as) —j^- For example, if the mass of either

body be doubled, the product {inm'') of the masses is doubled,
and consequently the attraction is doubled. If the distance

^2 4/

between their centers of mass be doubled, then ( —

the attraction becomes one-fourth as great.

97. Galileo^ s exjoeriment. — Galileo let fall from the top of
the leaning tower at Pisa iron balls of different masses, and
found that they fell with equal acceleration and reached the
ground at the same instant. This celebrated experiment
established two important facts : —

(1) At any given place the acceleration due to gravitation is
independent of the mass of the falling body ; in other words,

1 " It may turn out to be a property inherent in matter, and an exception to every
known case ; but it is more probable that it is not an inherent property in matter at
all, but a property due to a strain in the medium in which all matter is immersed."
(Lodge.) The gravitation stress existing in this medium may be, in some respects,
analogous to the stress which exists in a stretched rubber band, which would tend to
bring together any bodies between which it might be stretched.

2 The attraction between two gram-masses whose centroids are one centimeter
apart is one fifteen millionth of a dyne.

120 MOLAR DYNAMICS.

for all bodies at the same place the acceleration due to the
earth's attraction is the same.

Let / and f be the intensities of two attractive forces
acting to move two bodies, whose masses are respect-
ively m and m\ to the earth ; then (p. 40)

f=m a, and f = m a^
but, as proved by Galileo's experiment,
a = a,

hence (diviam^) ^ = -^ ,

^ ^ f m

and, in general, f^m,

i.e. (2) the intensity of the earth^s attraction at the

same place varies as the mass.

In other words, the deductions from this experi-
FiG. 87. j^rigj^^ r^pe . ^^ ^jia^i^ 2i\\ free bodies, whatever their
mass, fall toward the earth with equal accelerations, and
(2) that if one body possess twice the mass of another, twice
the force is required to give it the same acceleration.

Proposition (1) is seemingly contradicted by every-day
experience, for if a coin and a piece of tissue paper be
dropped from a hight they fall with very different velocities
and accelerations. But if a coin and several bits of paper
be placed in a long glass tube (Fig. 87), the air exhausted,
and the tube turned end for end, it will be found that the
coin and the papers fall in the vacuum with equal velocities.
It is evident, then, that when there is a difference in the
acceleration of falling bodies at the same place it is not due
to the force of gravitation but to some other force, e.g. the
resistance of the air.

98. Variation of gravitation^ or g, on the earth'' s surface. —
A spherical body of uniform density acts upon a particle out-
side it as if the entire mass were collected at its center. If
the earth were a homogeneous sphere and at rest, then the

WEIGHT ABOVE THE EARTH's SURFACE. 121

value of g would be constant at its surface since every point
in it would be equidistant from the center. But the earth
is a spheroid, its polar diameter being about 43 kilometers
(nearly 27 miles) less than the equatorial diameter. Conse-
quently the value of g is less in the equatorial than in the
polar regions, i.e. a given body stretches a spring balance less
as it is carried from either pole toward the equator. 1[?he loss
of weight of any body due to this increase of distance from
the center of the earth in being transported from the poles to
the equator, is estimated to be ^\^ of its weight at the poles.
But we have previously seen (p. 74) that the centrifugal
force at the equator diminishes the weight of a body -^\^.
Now in consequence of difference in distance from the center
of mass of the earth and difference in velocity due to the earth's
rotation, a body weighs at the equator s^g + 2^9 = \\^ less
than at the poles.

99. Weight above the earWs surface. — We infer from the
law of gravitation that a body weighs more at the earth's
surface than above it ; in other words, bodies become lighter
as they are raised above the earth's surface. But since the
force diminishes as the square of the distance from the center
(not from the surface) of the earth, and as the surface is about
4,000 miles from the center, the diminution for a few miles or
for any distance which we are able to raise bodies is scarcely
perceptible ; hence in all commercial transactions we may,
without important error, buy and sell as if the weighing
always took place at the same distance from the center of
the earth, in which case mass is strictly proportional to
weight.

100. Weight helow the eartKs surface.

It may be demonstrated geometrically (Wood's Elementary
Mechanics, p. 38, Barker's Physics, p. 117) that if a particle be
placed anywhere within a homogeneous hollow spherical shell of
matter, as at a, c, or d (Fig. 88), it is in a state of equilibrium in

122 MOLAR DYNAMICS.

regard to the attraction of the matter of the enveloping shell, i.e. a
body thus placed within a hollow shell of the earth, of whatever
thickness, would be under the influence of balanced forces and
hence weightless. Hence if a body be taken below the surface of
the earth, as from a to a' (Fig. 89), it is practically placed within a
hollow spherical shell of the earth, and therefore is freed virtually

Fig. Sd.

from the gravitation influence of this shell. Its weight is now
wholly determined by the gravitation of the smaller mass m n. So
that as a body is carried below the surface of the earth, it loses
in weight as much as it would if it were being transferred to
smaller and smaller earths ; consequently at the earth's center
matter is under the influence of balanced forces and hence weight-
less. If the earth were a perfect sphere and homogeneous, the
weight of a body would diminish uniformly as the distance from
the surface increased.

Exercises.

1. a. Which is independent of mass, weight or acceleration ? 6. Which
varies as the mass '?

2. Why does a hundred-pound iron ball fall with no greater accelera-
tion than a one-pound ball of the same material ?

3. a. Which falls with greater acceleration in the air, an iron ball or
a wax ball ? Why ? h. How would their accelerations compare in a
vacuum ? c. Is acceleration independent of kind of matter ?

4. If the earth's mass were doubled without any change of volume,
how would it affect your weight ?

5. On what principle may you determine that the mass of one body is
ten times the mass of another body ?

6. How many times must you increase the distance between the centers
of two bodies that their attraction may become one-fourth as great ?

WEIGHT BELOW THE EARTH S SURFACE.

123

7. If a body on the surface of the earth be 4,000 miles from the cen-
troid of the earth, and weigh at this place 100 pounds, what would the
same body weigh if it were taken 4,000 miles above the earth's surface ?

8. The masses of the planets Mercury, Venus, Earth, and Mars are
respectively very nearly as 7, 79, 100, and 12 ; assuming that the distance
between the centers of the first two is the same as the distance between
the centers of the last two, how would the attraction between the first
two compare with the attraction between the last two ?

9. What would be the answer to the last question
if the distance between the centers of the first two
were four times the distance between the centers of
the last two ?

10. Would the weight of a soldier's knapsack be
sensibly less if it were carried on the top of his rifle ?

11. If you hold a body on a spring-balance in an
elevator, what effect will be noticed as you start to
ascend ? What effect, as you start to descend ? Ex-
plain. '

12. Let E (Fig. 90) represent the earth as a perfect
homogeneous sphere of a radius of 4,000 miles, a. If
a body at a weigh 1 pound, what would it weigh at ?n,
1,000 miles below its surface ? h. What at ?2, o, and
c, respectively 2,000, 3,000, and 4,000 miles below the
surface ? c. What at 6, d, e, and i, respectively 4,000,
8,000, 12,000, and 2,000 miles above the earth's sur-
face ?

13. a. What is a vertical line ? 6. What angle does it make with the
frame of a spirit level when in position ?

14. A body weighs 100 pounds at the earth's surface. At what two
places would its weight on a spring balance be 50 pounds ?

15. If the acceleration at sea-level be 32.2 feet, what is it 5 miles above
sea-level ?

Fig. 90.

124 MOLAR DYNAMICS.

CHAPTER III.

PROPERTIES OF MATTER.

Section I.

CONSTITUTION OF MATTER.

101. Minuteness of idarticles of matter. The molecule. —
Physiology teaches us that, in order to smell any substance,
we must take into our nostrils, as we breathe, small particles
of that substance which are floating in the air. The air, for
several meters around, is sometimes filled with fragrance from
a rose. You cannot see anything in the air, but it is, never-
theless, filled with a very fine dust that floats away from the
rose. At sea the odor of rosemary renders the shores of
Spain distinguishable long before they are in sight. A grain
of musk will scent a room for many years, by constantly
sending forth into the air a dust of musk. Though the
number of particles that escape must be countless, yet they
are so small that the original grain does not lose perceptibly
in weight.

These instances, and numerous others of common obser-
vation, give us only a feeble conception of the minuteness
of particles of matter and of its extreme divisibility. Yet
the smallest particle of dust of rosemary or musk, and the
smallest particle of any substance which can be obtained by
any mechanical means, is very large in comparison with
bodies called molecules, which, of course, are too small to be
seen, but of whose existence we have ample evidence. A
single simple example of the proofs of their existence, though
by no means the most conclusive, must suffice at this place.

THEORY OF THE CONSTITUTION OF MATTER. 125

Matter, e.g. gold or water, is either continuous as it appears
to the eye or it is discontinuous, granular, composed of distinct

particles (called molecules) somewhat as rep- ^ ^

resented in Fig. 91. Matter is compressible !-':'.-*'' .*••'•'.

and expansible. On the supposition that ^'"rr^TE" ■.'•''••■"■• ■.••*•
matter is continuous, these properties are •.'••'.•;••••.•'•'.

unexplainable ; but on the supposition that " ':..•..'. v.-.-.;' "'

matter is molecular, these properties are '^°''Ita?I" ;};•;•'; •;".U-:;
easily explainable. A change of volume by •••■•■•■•■■■••••■.■■.

contraction or expansion means simply a
coming together or a separation of the molecules composing
the body, as represented in Fig. 91.

102. Theory of the constitution of matter. — For reasons
which will appear as our knowledge of matter is extended,
physicists have generally adopted the following theory of the
constitution of matter : Every body of matter except the mole-
cule is composed of exceedingly small disconnected particles,
called molecules. No two molecules of matter in the universe
are in permanent contact with each other. Every molecule is in
quivering motion, Tnoving back and forth between its neighbors,
hitting and rebounding from them. When we heat a body we
simply cause the molecules to move more rapidly through their
spaces; so they strike harder blows on their neighbors, and
usually pjush them away a very little ; hence the body expands.

103. Porosity. — If the molecules of a body are never in
contact except at the instants of collision, it follows that
there are spaces between them. These spaces are called ijores.

All matter is porous ; thus water may be forced through the
pores of cast iron ; and gold, one of the densest of substances,
absorbs liquid mercury.

Impenetrability^ may be affirmed of molecules, but not

iThe doctrine of impenetrability declares that " Two bodies of matter cannot
occupy tbe same space at the same time." In its strict scientific sense this doctrine
is as axiomatic as the statement that " A body cannot be in two places at the same
time."

126 MOLAR DYNAMICS.

necessarily of masses. The term pore, in physics, is restricted
to the invisible spaces that separate molecules. The cavities
that may be seen in a sponge are not pores, but holes ; they
are no more entitled to be called pores, than the cells of a
honeycomb or the rooms of a house are entitled to be called,
respectively, the pores of the honeycomb or of the house.

By means of delicate calculations, physicists ascertain approxi-
mately the probable size of the molecule. Lord Kelvin estimates
that the diameter of the molecules of a gas cannot be less than one
five-hundred-millionth of a centimeter. The minimum particle
visible to the eye is a cube one four-thousandth of a millimeter on a
side. Such a cube contains from sixty to one hundred million
molecules.

"The kinetic theory of gases (p. 271) teaches that in a cubic inch

of any gas at atmospheric pressure and at ordinary temperatures

there are about 3 X lO^o detached particles absolutely similar and

' equal to one another. Here v^e reach the limit of our present

knowledge as to division of matter." (Tait.)

104. Atomic theory of imatter. Atoms.

The theory given above assumes that the molecule is tlie limiting
particle of possible physical division, i.e. the smallest particle of any
substance which can preserve the properties of that substance ;
hence the molecule is sometimes termed the "physicist's unit."
The chemist finds it necessary to assume that the molecule is capable
of a still further subdivision, a division which results in a complete
change in the character of the substance operated on. Thus a
molecule of sugar when subjected to chemical processes, which are
virtually chemical divisions, yields carbon, hydrogen, and oxygen,
substances entirely unlike sugar. These still smaller particles
obtained by the division of the molecule are called atoms. The
atomic theory assumes that the atom, as the word etymologically
signifies, is indivisible, and it may be termed the " chemist's unit."
An atom is indestructible and unchangeable. About seventy differ-
ent kinds of atoms have been discovered. These constitute the
so-called elementary substances. All other substances are compounds
of certain of these elements of varying degrees of complexity. A
molecule consists of a group of atoms bound together by chemical

THE PECULIAR PROPERTIES OF MATTER. 127

forces usually termed chemical affinity, and a mass consists of a
group of molecules which may or may not be bound together by a
physical or molecular force called cohesion.

Section II.

THE STATES OF MATTER AND THEIR PECULIAR PROPERTIES.

105. Solids, liqicids, and gases. — In popular language there
are said to be three states of matter, the solid, the liquid, and
the gaseous.

Solids preserve a definite volume and shape when left to
themselves ; liquids tend to preserve a definite volume only,
while their shape conforms to that of the containing vessel ;
gases tend to preserve neither a definite volume nor shape, but
conform not only in shape but in volume to the containing
vessel, no matter how large this may be. We may have a
vessel half full of liquid, but a mass of gas always occupies
the whole of a containing vessel, however small the quantity
of gas. Gases tend to expand indefinitely and to assume an
infinite volume with a correspondingly small density. Solids
and liquids may have free surfaces, gases cannot retain per-
manently a free bounding surface independent of the contain-
ing vesseL

Which of the three states any portion of matter assumes depends
upon its temperature and pressure. Just as at ordinary pressures of
the atmosphere water is a solid {i.e. ice), a liquid, or a gas {i.e. steam),
according to its temperature, so any substance may be made to
assume any one of these forms unless a change of temperature
causes a chemical change, i. e. causes it to break up into other sub-
stances. Eor example, wood cannot be melted, because it breaks
up into charcoal, steam, etc., before the melting-point is reached.
In order that matter may exist in a liquid (and sometimes in a solid)
state, a certain definite pressure is required. Ice vaporizes, but
does not melt {i. e. liquefy) in a space from which the air (and con-
sequently atmospheric pressure) has been removed. Solid carbonic

128 MOLAR DYNAMICS.

acid vaporizes, but does not melt unless the pressure is greater
than the ordinary atmospheric pressure. Charcoal has been vapor-
ized, but has never been liquefied, undoubtedly because sufficient
pressure has never been used.

As regards the temperature and pressure at v^hich different sub-
stances assume the different states, there is great diversity. Oxygen
and nitrogen gases liquefy and solidify only at extremely low tem-
peratures ; and then, only under great pressure. On the other
hand, certain substances, as quartz and lime, are liquefied only
by the most intense heat.

106. Fluids. — The term implies the property of flowing.
Since both liquids and gases possess this property in an
eminent degree in consequence of great freedom of motion of
their molecules around one another, they are both included
under the common term fluid. Further on it will be seen that
one of the chief distinctions between a solid and a fluid is
that the former possesses rigidity, while the latter does not.

107. Vapo7'ous state ; critical state.

Closer study makes evident that the above classification is purely
r.rbitrary. The three states of matter sometimes merge into one
another so that there remains no distinct line of demarkation
betvt^een them. For example there is a state which may be regarded
as intermediate between the liquid and the gaseous, called the
vaporous state. If a substance in the gaseous form be compressed
or cooled to such an extent that it will suffer but little further
compression or cooling without passing into the liquid state, it
possesses in this state certain peculiar properties and is known by
the name of vapor. A vapor may be defined as a gas near its con-
densing point. "When matter is at a temperature and under a pres-
sure such that if heated a little more it becomes a vapor, or if
allowed to cool a little more it becomes a liquid, it is said to be in
the critical state. In this semi-liquid state the gaseous and liquid
states meet and are indistinguishable. The highest temperature
at which this occurs is called the critical temperature, and the
highest pressure the critical pressure. A vapor may be defined, also,
as any gaseous substance at a temperature below its critical temper-
ature.

MOLECULAR FORCES. 129

For example, carbon dioxide at 31° and under a pressure of 73
atmospheres is in a critical state. Heated a little it certainly becomes
gaseous ; cooled a little it as certainly assumes the properties of a
liquid, since it is far less compressible. But if the pressure be
maintained, the transition from one to the other is not recognizable.

108. Ultixigaseous or fourth state of matter.

Air has been rarefied to the three-hundred-millionth of its normal
density. But when gaseous matter is rarefied to even a millionth
of the density of air at sea-level, it exhibits extraordinary properties
quite as different from the gaseous as this is from the liquid state,
so that some are disposed to consider that there is an ultragaseous
or fourth state of matter.

Section III.

MOLECULAR FORCES.

109. Molecular attractive forces. — Many of the properties
of matter are due to molecular forces, some of wliicli now
demand our attention. For convenience we call bodies of
appreciable size molar (massive) bodies, or masses, in dis-
tinction from molecules (bodies of very small mass). Action
between molar bodies, usually at sensible distances, is called
molar force ; action between molecules, always at insensible
distances, is called m^olecular force. According to the theory
of the constitution of matter the molecules of every mass are
in ceaseless motion, hitting and rebounding from one another.
This tends to drive the molecules apart. In gaseous masses
the molecules move without restraint ; hence gaseous bodies
always tend to expand.

In solids and liquids the molecules are held under the
action of a very powerful attractive force, called cohesion,
which prevents their separation except under the action of
considerable external force. It is the force which resists an
effort tending to break, tear, or crush a body. The tenacity or

130 MOLAR DYNAMICS.

tensile strength of solids and liquids, i.e. the resistance which
they offer to being pulled apart, is due to this force. It is
usually greater in solids than in liquids, and is entirely
wanting in a true gas.

110. Strain, rigidity, elasticity. — Strain means change of
size, change of shape, or deformation of any kind. Change-
of-size strain is called compression or dilatation, and the
resistance of matter to it is called elasticity of volume.
Change-of-shape strain, such as in flexion, torsion, etc., is
called distortion, and the resistance to it manifests itself
either as elasticity of figure, or rigidity.

Elasticity is that property in vii'tue of ivhich a solid tends to
recover its size and shape, and a fluid its size, after defor-
mation. Solids are remarkable for high rigidity. A per-
fectly rigid solid is one which, when a force is applied to it
in any way, suffers no strain before breaking. No body is
absolutely rigid, though some bodies are approximately so.
If the stress between the molecules in opposition to the
distorting force continue constant, regardless of the time the
strain is kept up, and restore the body to its normal con-
dition immediately on the removal of the distorting force,
without any permanent strain or "set," the body is said to
be perfectly elastic. All fluids are perfectly elastic, and a
few solids are approximately so, such as ivory, steel, and
glass.

If a solid have little or no tendency to recover its size and
shape after distortion, it is said to be plastic or inelastic.^
Such substances are putty, wet clay, and dough. A great
number of substances are elastic when the distorting forces
are small, but break or receive a " set " when these forces
are too great. They are said to be elastic '^ within certain
limits," called the limits of elasticity. If strained beyond
those limits, they become more or less plastic. Hence the
springs of a buggy sometimes become set from bearing a too

VISCOSITY. 131

heavy load and lose permanently mucli of their elasticity ;
i.e. they become in a degree plastic.
111. Viscosity.

Experi77ient 1. — Support in a horizontal position, by one of its extrem-
ities, a stick of sealing-wax, and suspend from its free extremity an
ounce weight, and let it remain in this condition several days, or perhaps
weeks. At the end of the time the stick will be found permanently bent.
Had an attempt been made to bend the stick quickly, it would have been
found quite brittle.

It may seem like an abuse of the term to call sealing-wax
a fluid, yet the experiment shows it to be a fluid, or at least
to possess fluidity, or freedom of motion of its molecules
around one another, in a small degree. Eesistance to deforma-
tion due to the friction of the molecules of a body in sliding
over one another is called viscosity. Bodies that slowly suffer
continuous and permanent deformation under the action of a
continuous stress are said to be viscous. A lump of pitch in
course of time loses its sharpness of outline and flows down
hill of its own weight. It is very viscous. Cold molasses is
quite viscous, but as its temperature is raised its viscosity
diminishes and it becomes more and more plastic or mobile.
A perfectly rigid solid is one of infinite viscosity. A jperfect
fluid is a fluid which possesses no viscosity. Gases are
viscous to some extent and are therefore imperfect fluids.

Bodies surrounded by air have on their surfaces an adherent film
. of air. When they move, this film rubs against the surrounding air,
and thus their movements are retarded by friction in the air. To
the viscosity of the air is due in part the retardation of the velocity
of falling bodies. A penny and a piece of tissue paper fall with
equal accelerations in a vacuum (p. 120), but in the air the penny
falls more rapidly because it presents less surface and therefore is
retarded less in proportion to its weight by the friction of the air.

Falling water is retarded by the air ; conversely, air is dragged
down by falling water, as shown by the following experiment.

132

MOLAR DYNAMICS.

Experiment 2. — Take a long glass tube A (Fig, 92), funnel-shaped
at one end, and having a little below this end
a short branch tube a. Pour water into the
funnel and observe that it falls in a con-
tinuous stream until it reaches a, but below
this point it is broken up by descending
bubbles of air which it drags along with it
after coming in contact with the air at a.
K direct sunlight be allowed to strike the
tube, the interior reflection from the water
in this part of the tube is dazzling and
beautiful.

Connect with a by means of a rubber
tube another short piece of glass tube 6, and
introduce the latter into a tumbler of water.
The water descending in A exhausts the air
in 6 and the outside atmospheric pressure
causes the water in the tumbler to ascend
this tube 6, and thus the tumbler may be
emptied. If a closed vessel B be connected with a, a high vacuum
may be obtained in it.

Fig. 92.

Experiment 3. — Pour water from an elevation upon a still body of
water below and observe the bubbles of air which form in the water, the
air being dragged by the falling current to a considerable depth in the
liquid.

112. Hardness. — Hardness is resistance to abrasion or
scratching.

To enable ns to express degrees of hardness, the following
table of reference is generally adopted : —

MOHR'S SCALE OF HARDNESS.

1. Talc.

2. Gypsum (or Rock-Salt).

3. Calcite.

4. riuor-Spar.

5. Apatite.

10.

Orthoclase (Feldspar).

Quartz.

Topaz.

Corundum.

Diamond.

CAPILLARITY. 133

By comparing a given substance with the substances in the
table, its degree of hardness can be indicated approximately.
Thus '-11 = 7" means that the body is about as hard as quartz.

113. Malleability, ductility. — Solids which possess that
kind of fluidity which renders them susceptible of being rolled
or hammered out into sheets are said to be malleable. Most
metals are highly malleable. Gold may be hammered so thin
as to be transparent, or to a thickness of one three-hundred-
thousandth of an inch. Most substances that are malleable
are susceptible also of being drawn out into fine threads,
e.g. wire of different metals. Such substances are said to be
ductile. Platinum has been drawn into wire .000165 inch
thick, or so fine as to be scarcely visible to the unaided eye.

Section IV.

CAPILLARITY.

114. Cohesion of liquids. — Clean glass is wet by water.
If a glass plate be dipped into water and then withdrawn, a
layer of water clings to the glass. When the glass is with-
drawn, water is torn from water, and not glass from water.
This shows that the attraction of the molecules of water for
one another is weaker than the attraction between glass and
water. Or if, to save words, we call the attraction between
the solid and the liquid adhesion, then we may say that the
cohesion between the molecules of the water is weaker than
the adhesion between the glass and the water.

Clean glass is not wet by clean mercury, which shows that
the adhesion between glass and mercury is not as great (about
one third as great) as the cohesion in mercury. Generally
speaking a solid is wet by a liquid when the adhesion of the
solid to the liquid is greater than the cohesion of the liquid,
and is not wet when the cohesion is greater than the adhesion.

134 MOLAE DYNAMICS.

115. Surface tension. — When a rubber band is strained or
stretched, it is said to be in a state of tension, and there exists
between its molecules a contractile or resilient stress.

In liquids the molecules are within the limits of one another's
attractions, which accounts for a greater or less viscosity or
hindrance of flow and also for a certain phenomenon called
surface tension. Every liquid behaves as if a thin film forming
its external layer were in a state of tension, or were exerting
a constant effort to contract.

It is not within the scope of this book^ to explain in full
the dynamics of the molecular forces by which this result is
brought about ; it must suffice to call the attention of the
student to the peculiar condition, with reference to mutual
attractions, of those molecules which compose the surface film.
In the interior of a liquid each molecule is surrounded by
other similar molecules and the position which it assumes is
that in which it is acted upon equally in all directions, and
there is nothing to render the mutual attractions manifest.
At a free surface, however, the molecules can be acted upon
only by others lying internal to them. The result is a system
of forces acting at right angles to the free surface of the
liquid, and tending to reduce the free surface to the least
possible area. This tendency of a liquid surface to contraction
means that it acts like an elastic membrane, equally stretched
in all directions, and by a constant tension. In the case of
pure water at 20° C. this tension is about 81 dynes per linear
centimeter.

Experiment. — Form a soap-bubble at the orifice of the bowl of a
tobacco pipe, and then, removing the mouth from the pipe, observe that
the tension of the two surfaces (exterior and interior) of the bubble drives
out the air from the interior and finally the bubble contracts to a flat
sheet of minimum area.

^ The student who is desirous of knowing more of this interesting subject may
consult the article " Capillarity," by Maxwell, in the Encyclopsedia Britannica.

CAPILLAKY PHEXOMEXA. 135

As a consequence of surface tension, eyery body of liquid tends to
assume the spherical form, since the sphere has less surface than any-
other form having equal volume. In large bodies the distorting forces
due to gravity are generally sufficient to disguise the effect ; but in small
bodies, as in drops of water or mercury, it is apparent. Again, if the
distorting effect of weight be eliminated in any way, as by immersing a
quantity of oil in a mixture of water and alcohol of its ovrai density, or by
replacing the central portion of the body by a fluid much lighter than its
own kind, as in the case of a soap-bubble, the sphere is the resulting form.

116. Cajjillari/ 2yhenonie7ia. — Surface tension is by no means
peculiar to liquids. The surfaces of all bodies tend to con-
tract. But since gases have no surfaces of their own, and the
rigidity of solids prevents an alteration of shape, it is obvious
why liquids show the effects of surface tension most readily.
But the surface tensions of solids and gases perform their
part in determining certain phenomena. For example, if a
glass rod be thrust vertically into water so as to leave a jjart
projecting into the air, the surface of the water does not meet
the rod at right-angles, but is turned up so as to form a very
small angle ^ with the surface of the glass, as acb (Fig. 93).
Here the three substances, water, glass, and air, are brought
in contact and there are a triplet of tensions in operation the
resultant of which is a force which pulls the water up against
the glass wall. On the other hand if mercury, glass, and air
be brought in contact, the relation between the triplet of
forces becomes so changed as to cause the mercury to meet the
glass at a- very large angle, about 135°. It thus seems that
when a solid, a liquid, and a gas are in contact, their boundary
surfaces form contact angles with one another determined by
their relative surface tensions.

If a glass tube x (Fig. 94) of capillary (hair-like) bore be
thrust into water, the water will rise in the bore considerably
above the general level outside. If a similar tube y (Fig. 95)

1 If the glass be quite clean tlie angle is 0. If not clean, it may reach, and even
exceed, 90°. •

136

MOLAR DYNAMICS.

be thrust into mercury, the mercury within the bore will be
depressed below the surface outside. Phenomena of this kind
are called capillary idhenomena. The surfaces of the liquids
inside the bores are curved, the surface of water being concave
and that of mercury convex. The size of the bore of the

Water
Fig. 94.

"til

a A i 1 e

Fig. 96.

tubes X and y is greatly exaggerated in order to show this
more plainly. The concavity and convexity of these interior
surfaces are a necessary consequence of the angles of contact
with which these liquids meet glass. It remains only to
explain the elevation and depression of the column of liquid
in the tube. This may be done in part by analogy. Let AB
(Fig. 96) represent a clothes line suspended slackly between
two posts. From this line hang by strings small stones
a, h^ c, etc. If the hempen line become wet, as in a rain, it

CAPILLARY PHENOMENA.

137

contracts and straightens, as shown by the dotted line AB.
In other words, the contractile force which is exerted obliquely
{e.g. nm Fig. 96) is resolvable into two forces, one of which
is horizontal and the other is vertically upward ; the latter
tends to elevate the stones. In a similar manner the curved
surfaces of water and mercury tend to contract and become
flat. In the case of the water surface (which is concave) the
contractile force tends to elevate the pendent liquid ; but in
the case of the mercury surface (which is convex) the tendency
is to produce depression. On the nature of the curvature
depends the direction in which the contractile force acts on
the pendent liquid. Now it is evident that water will be
drawn up by this contractile force until the weight of the
column balances this force ; and mercury will be depressed
until the force is balanced by the pressure of the mercury out-
side the tube. Capillary phenomena are, therefore, phenomena
of surface tension.

The phenomena of capillary action are well shown by placing
various liquids in U-shaped glass tubes having one arm
reduced to a capillary size, as A
and B in Fig. 97. Mercury poured
into A assumes convex surfaces
in both arms, but does not rise as
high in the small arm as it stands
in the large arm. Pour water into
B, and all the phenomena are re-
versed. Fig. 98 shows the forms
that the surfaces of water and
mercury take when contained in the same glass tube.

The following laws of capillary action may be verified by
experiment : —

I. Liquids rise in tubes tvhen they wet them, and are de^jressed
when they do not.

Fig. 97.

Fig.

138 MOLAR DYNAMICS.

II. The elevation or depression varies inversely as the diameter
of the bore.

III. The elevation and depression vary with the nature of the
liquids emjjloyed, and with the substance of the tube.

lY. The elevation or depression varies inversely ivith the tem-
perature.

Section V.

DIFFUSION OF FLUIDS.

117. Other molecular phenomena.

Besides the phenomena we have just studied, there are a great
many others depending in part on molecular attraction, but much
more on molecular motions, of which we learned on p. 125 and
which we now must consider more in detail.

The molecules of all bodies are constantly in a state of motion.
The higher the temperature the greater are their velocities. In a
solid "a molecule, though in continual motion, never gets beyond
a certain very small distance from its original position in the body.
In fluids there is no restriction to the excursions of a molecule.
True the molecule travels only a very small distance before it
encounters another molecule ; but after this encounter there is
nothing which determines the molecule rather to return towards the
place whence it came than to push its way into new regions. Hence
in fluids the path of a molecule is not confined to a limited region,
but may penetrate to any part of the space occupied by the fluid."
Maxwell.

118. Diffusion of liquids.

Experiment 1. — Partially fill a glass jar (Fig. 99) with water.
Then introduce beneath the water, by means of a long tunnel, a
concentrated solution of sulphate of copper. The lighter liquid
rests upon the heavier, and the line of separation between the two
liquids is at first distinctly marked. But in the course of days or
weeks this line will gradually become obliterated, the heavier blue
liquid will gradually rise, and the lighter colorless liquid will
descend, till they become thoroughly mixed.

Experiment 2. — Take about 1 cc of bisulphide of carbon, color
it by dropping into it a small particle of iodine, and pour this

DIFFUSION OF GASES.

139

colored solution into a test-tube nearly filled with water. The

colored liquid, being heavier than the water, sinks

directly to the bottom, and . shows no tendency to

mix with the water. But in the course of time you

discover that the colored liquid diminishes in quantity,

and finally disappears. The peculiar odor of this

substance which pervades the air in the vicinity shows

that a considerable portion has evaporated. But it

must have worked its way gradually through the

water above it.

If, in the last two experiments, you examine the
liquid with a microscope during the operation, you
will not be able to trace any currents ; hence the motion of the liquids
is not m mass, but by molecules — a true intermolecular motion.

An intermingling of the molecules of two liquids caused by their
own motions is called diffusion of liquids.

119. Diffusion of gases.

Experiment 3. — Fill a test-tube with oxygen gas, and thrust into
it a lighted splinter ; the splinter burns much more rapidly than in
the air. Fill another tube with hydrogen gas, and keep
the tube inverted (for, this gas being about 14.4 times
as light as air, there will be no danger of its falling out).
Thrust in a lighted splinter ; the gas takes fire, and burns
with a pale flame at the mouth of the tube. Next fill
one tube with oxygen and the other with hydrogen
gas, and place the mouth of the latter over the mouth
of the former, as in Fig. 100. In about a minute apply
a lighted splinter to the mouth of the tube (let the mouth
of each tube be freely open to prevent accident) ; a slight
explosion takes place in each instance. It is apparent
that although the oxygen gas is 14.4 times as heavy as
the hydrogen, some of it has risen into the upper tube,
while some of the lighter , hydrogen has descended into
the lower tube, and the two gases have become diffused.

There are liquids between which diffusion does not
take place. But it does take place between any two
gases whenever they are placed in contact.

In consequence of this universal tendency to diffusion,

gases will not remain separated — i.e. a lighter resting

Fig. 100. upon a heavier, as oil rests upon water. This is of

140

MOLAR DYNAMICS.

immense importance in the economy of nature. The largest portion
of our atmosphere consists of a mixture of oxygen and nitrogen
gases. There are always present also small quantities of other gases,
such as carbon dioxide, ammonia gas, and various other gases,
which are generated by the decomposition of organic matter. These
gases, obedient to gravity alone, would arrange themselves according
to their weight, — carbonic-acid gas at the bottom, or next the earth,
followed respectively by oxygen, nitrogen, ammonia, and other gases.
Neither animal nor vegetable life could exist in this state of things.
But, in consequence of the diffusibility of these gases, they are
found intimately mixed and in the same relative proportions,
whether in the valley or on the highest mountain peak.

120. Osmose, or diffusion of liquids through membranous septa.

Certain liquids when separated from each other by membranous
septa diffuse through these septa, and the diffusion may be more
rapid than when no septa intervene. Such membranes as a bladder,
cow's pericardium, and parchment paper are especially suited to
this purpose. Diffusion through septa is called osmose. There is
a wide range of relative diffusibilities. At one end of the scale are
such substances as solutions of urea, common salt, and such sub-
stances as crystallize, hence called crystalloids. These diffuse
^ rapidly. At the other extreme are such substances as
starch, gum, albumen, gelatine, and glue-like matter,
called colloids. These diffuse very slowly. If a mixture
composed of a crystalloid and a colloid be placed in a
bladder, for instance, and the bladder be suspended in
water, the crystalloid will rapidly diffuse through the
septum while the colloid will diffuse very slowly. In
this way a separation of the two substances may be
eft'ected. Separation by this process is called dyalysis.

121. Osmose of gases.

Experiment 4. — Seal the open end of a thin, unglazed

earthen cup, such as is used in a Bunsen battery (p.

471), with plaster of Paris, through which extends a

glass tube. Place the exposed end of the tube in a

Fig. 101. cup of colored water. Lower a glass jar filled with

hydrogen or coal-gas over the porous cup, as in Fig. 101. Instantly

air is forced down through the tube, and escapes in bubbles from

OSMOSE OF GASES.

141

the colored liquid. The gas in the larger vessel forces its way
through the pores of the cup, diffuses through the air contained in
it, and causes an unusual pressure on the colored liquid, as is
evinced by the air that is forced out through it. In a minute remove
the glass jar. The hydrogen now escapes through the sides of the
cup, and mixes with the air on the outside ; a partial vacuum is
formed in the cup, and water rises in the tube. In each case air
passes through the sides of the porous cup, but the influx and efilux
of the hydrogen is much more rapid than that of the air.

An interesting modification of this apparatus is the diffusion
fountain (Fig. 102). By passing the glass tube of the porous cup
through the cork of a tightly-stoppered vessel,
and having another glass tube pass through
another perforation in the same cork, water is
forced out in a jet several feet in hight, when
the hydrogen jar is held over the porous cup.

Children well understand that toy balloons,
which are made of collodion and filled with
coal-gas, collapse in a few hours after they are
inflated. This is caused by the escape of the gas
by osmose. Nature furnishes an illustration of
osmose of gases in respiration. In the lungs
the blood is separated from the air by the thin,
membranous walls of the veins. Carbon dioxide
escapes from the blood through these septa, and oxygen gas enters
the blood through the same septa.

The phenomena of diffusion both in liquids and gases furnish strong
and tangible evidence that these bodies consist of molecules in a state
of continual motion.

Fig. 102.

142

MOLAR DYNAMICSo

CHAPTEE IV.

DYNAMICS OF FLUIDS.

Section I.

TRANSMISSION OF PRESSURE.

122. Laiu of hydrostatiG and 'pneumatic transmission of
jjressure. — That branch of science which treats of liquids in a
state of equilibrium or rest is called hydrostatics ; that branch
which treats of liquids in motion is called hydrokinetics ;
and that branch which treats of the dynamics of air and other
gases is called pneumatics. With the
exception of phenomena occasioned
by difference in compressibility and
expansibility, liquids and gases are
subject to the same laws and may
be treated togetlier, in so far as they
are alike, under the common term
fluid.

Experiment 1. — Fill the glass globe and
cylinder (Fig. 103) with water, and thrust
the piston into the cylinder. Jets of water
will be thrown not only from that aperture
a in the globe toward which the piston
moves and the pressure is exerted, but from .
all the apertures.

If a finger be placed loosely over
the end of a water faucet, spray will
be thrown to equal distances in all
directions. It thus appears not only
that external pressure is exerted
upon that portion of the liquid that lies in the path of the

Fig. 103.

TRANSMISSION OF PRESSURE. 143

force, but that it is transmitted equally to all parts and in all
directions.

When pressure is exerted upon a solid, on account of its
rigidity it is incapable of transmitting the pressure in other
than the direction in which it is pressed. With fluids it is
widely different. On account of the mobility of their mole-
cules, they are incapable of resisting a change of shape when
acted upon by a force which is not equally applied over the
whole surface of the body of fluid, hence any force applied
to a fluid body must be transmitted by the fluid in every
direction. Consequently every portion of the interior walls
of the containing vessel with which the fluid is in contact is
subjected to pressure.

Experiment 2. — Measure the diameter of the bore of each arm of the

glass U-tube (Fig. 104). We will suppose, for illustration, that the

diameters are respectively 40 mm and 10 mm ; then the ratio of the areas

of the transverse sections of the bores will be 40^ : 102=16 ; that is, when

the tube contains a liquid, the area of the free surface of the liquid in the

large arm will be 16 times as great as of that in the small arm. Pour

mercury into the tube until it stands about 1 cm

above the bottom of the large arm as cd. The

mercury stands at the same level in both arms.

Pour water upon the mercury in the large arm until

this arm lacks only about 1 cm of being full, as a b.

The pressure of the water causes the mercury to

rise in the small arm, and to be depressed in the

large arm. Pour water very slowly into the small

arm from a beaker having a narrow lip, until the

Fig 104
surfaces of the water in the two arms are at the

same level. It is evident that the quantity of water in the large arm is

16 times as great as that in the small arm.

This phenomenon appears paradoxical (cipparently contrary
to the natural course of things), until we master the important
hydrostatic principle involved. We must not regard the body
of mercury as serving as a balance beam between the two
bodies of water, for this would lead to the absurd conclusion

^g^

^■r ),

144

MOLAR DYNAMICS.

that a given mass of matter may balance another mass 16
times as great. We may best understand this phenomenon
by imagining the body of liquid in the large arm ^ to be
divided into cylindrical columns of liquid of the same size as
that in the small arm. There will evidently be 16 such
columns. Then whatever pressure is exerted on the mercury

Fig. 105.

by the water in the small arm is transmitted by the mercury
to each of the 16 columns, so that each column receives an
upward pressure, or a supporting force equal to the weight of
the water in the small arm.

The pressure exerted by a fluid upon the vessel containing
it is normal to the walls of the vessel. Fluid pressure is
expressed by stating the force exerted on a unit area, as 2 lbs.
per sq. in., 5 g per cm^, etc. The total pressure on any surface

TRANSMISSION OF PRESSURE.

145

is the product of the pressure per unit area multiplied by the
number of units of area.

Experiment 3. — Fig. 105 represents a section of an apparatus called
(from the number of uses to which it may be put) the seven-in-one appa-
ratus. A is a hollow cylinder closed at one end. B is a tightly fitting
piston which may be pushed into or drawn out of the cylinder by the
handle C when screwed into the piston. D is another handle permanently
connected with the closed end of the cylinder. E is a nipple, opening
into the space below the piston. To this may be attached a thick- walled
rubber tube F. G is a stop-cock and H is a funnel, either of which may
be inserted at will into the free end of the tube.

Support the seven-in-one apparatus with the open end upward, force
the piston in, place on it a block of wood A (Fig. 106), and on the block
a heavy weight. Attach one end of the rubber tube
B (12 feet long) to the apparatus, and insert a funnel
C in the other end of the tube. Raise the latter
end as high as practicable, and pour water into
the tube. Explain how the few ounces of water
standing in the tube can exert a pressure of many
pounds on the piston, and cause it to rise together
with the burden that is on it.

Fig. 106.

Fig. 107.

Experiment 4. — Remove the water from the apparatus, place on the
piston a 16-pound weight, and blow (Fig. 107) from the lungs into the
apparatus. Notwithstanding that the actual pushing force exerted
through the tube by the lungs probably does not exceed a few ounces,
the slight increase of pressure caused thereby, when exerted upon the
(about) 26 square inches of surface of the piston, causes it to rise together
with its burden.

146

MOLAR DYNAMICS.

A pressure exerted on a fluid enclosed in a vessel is trans-
mitted undiminished to every part of that vessel ; and the total
pressure exerted on the interior of the vessel is equal to the area
multiplied hy the pressure per unit of area.

123. The hydraulic press. — Closely allied to the seven-in-
one apparatus is the hydraulic press. It contains two pistons,
t and s (Fig. 108). The area of the lower surface of t is (say)
one hundred times that of the lower surface of s. As the
piston s is raised and depressed, water is pumped up from
the cistern A, is forced into the cylinder x, and exerts an

upward pressure against the
piston t one hundred times
greater than the downward
pressure exerted upon s.
Thus, if a pressure of one
hundred pounds is applied
at s, the cotton bales will be
subjected to a pressure of
five tons.

The pressure that may be
exerted by these presses is
enormous. The hand of a
child can break a strong iron
bar. But observe that, al-
though the pressure exerted
is very great, the upward movement of the piston t is very
slow. In order that the piston t may rise 1 cm, the piston s
must descend 100 cm. The disadvantage arising from slowness
of operation is insignificant, however, when we consider the
great advantage accruing from the fact that one man can
produce as great a pressure with the press as a hundred men
can exert without it.

The press is used for compressing cotton, hay, etc., into
bales, and for extracting oil from seeds. The modern engineer

Fig. 108.

PRESSURE OF FLUID DUE TO . ITS WEIGHT. 147

finds it a most efficient machine whenever great resistances
are to be moved through short distances.

124. Fi^essure of fluid due to its iveight. — Fluids exert
pressure due to their weight. Imagine a vessel filled with
shot ; you will understand that the upper layer of shot will
press upon the layer next beneath with a force equal to its
weight, the second upon the third with a force equal to the
sum of the weights of the first two, and so on. You will also
readily conclude that the pressure exerted upon the successive
layers will be exactly proijortional to their d&pths^ unless in
consequence of the great pressure to which the lowest layers
are subjected there should be a crowding together of the shot
so as to make them more compact. In this case there would
be a slight variation from the rule as stated. For a like reason
the downward pressure in a body of liquid increases as its
dejjth except in so far as the pressure is modified in conse-
quence of the compressibility of liquids. Liquids are, how-
ever, so slightly compressible that any variation in conse-
quence of the compression is usually neglected, and the
principle is stated in general that pressure at any point in a
liquid varies as its depth.

Since the shot possess a certain degree of mobility or free-
dom of motion around one another, their weight will cause to
some extent a lateral pressure against one another and against
the walls of the con- __-=_

taining vessel. In
consequence of the

extreme mobility of . ^Q^^^6^=^-r^=^-f^^=^
the molecules of fluids
the downward pres-
sure due to gravita-
tion at any point in fig. 109.
a fluid gives rise to an equal pressure at that point in all
directions. Hence the so-called PascaVs principle: At any
point in a fluid at rest the pressure is equal in all directions.

148

MOLAR DYNAMICS.

Thus, let a, h, c, etc. (Fig. 109), represent imaginary surfaces,
and the arrow-heads the direction of pressure exerted at
points in these surfaces at equal depths in a liquid. The
pressures exerted at these several points are equal.

The truth of this principle is obvious, for if there be any
inequality of pressure at any point, the unbalanced force will
cause particles at that point to move, which is contrary to the
supposition that the fluid is at rest. Conversely, when there
is motion in a body of fluid it is evidence of an inequality
of pressure.

125. Methods of calculating liquid pressure. — Conceive of

a square prism of water (Fig. 110), in the midst of a

body of water, its upper surface coinciding with the free

.^^^ m^^ ^^=^^ ^^^ - surface of the liquid. Let

^^g ^^^ ^ p^^^ ^ y" the prism be 4 cm deep and

j^M " "' ^^y " 1 cm square at the end ; then

the area of one of its ends is
1 cm^ and the volume of the
prism is 4 cc. Now the weight
of 4 cc of water is 4 g, hence
this prism must exert a
downward pressure of 4 g
upon an area of 1 cm^ But
at the same depth the pres-
sure in all directions is the
same, hence, generally, the
pressure at any depth in
water may be taken approxi-
mately as one gram per
square centimeter for each
centimeter of depth (=c= 1,000 k per m^ for each meter of
depth; or, since the weight of water is about 62.3 lbs. per
cu. ft., the pressure is 62.3 lbs. per square foot for each foot
of depth). In any other liquid, to determine the pressure at

Fig. 110.

METHODS OF CALCULATING LIQUID PRESSURE. 149

any depth the water pressure at the given depth must be
multiplied by the specific density (p. 177) of the liquid.

The conclusions arrived at may be summarized as follows :
In a mass of liquid at rest, the pressure is the same at all ^points
in any horizontal plane, and is equal to the weight of a column
of the liquid one square centimeter in section extending vertically
from the horizontal plane to a horizontal plane coinciding with
the upper surface of the liquid.

Experiment 5. — Take a square prism of pine wood 1 cm square at
its ends. Find its weight in grams and calculate to what depth it would
sink endwise in water. Take a test tube a little larger and longer than
the prism. Nearly fill the tube with water, lower the prism into the
water, and verify your calculation. How great is the upward pressure of
the water upon the bottom of the prism ?

Fig. 112.

Fig. 113.

Fig. 114.

Experiment 6. — A and B (Fig. Ill) are two bottomless vessels which
can be alternately screwed to a supporting ring C (Fig. 112). The ring
is itself fastened by means of a clamp to the rim of a wooden waterpail.

150 MOLAR DYNAMICS.

A circular disk of metal D is supported by a rod connected with one
arm of the balance-beam E. When the weight F is applied to the other
arm of the beam, the disk D is drawn up against the ring so as to supply
a bottom for the vessel above. Take first the vessel A, screw it to the
ring and apply the weight to the beam as in Fig.' 114. Pour water slowly
into the vessel, moving the index a up the rod so as to keep it just at the
surface of the. water, until the downward pressure of the water upon the
bottom tilts the beam, and pushes the bottom down from the ring, and
allows some of the water to fall mto the pail. Eemove vessel A, and
attach B to the ring as in Fig. 113. Pour water as before into vessel B ;
when the surface of the water reaches the index a, the bottom is forced
off as before.

That is, the pressures upon the bottoms of all vessels {of whatever
capacity or shape) are the same, provided the bottoms be of the same area
and the depth and density of the liquid be the same.

126. Rules for calculating liquid j^ressure agaiiist the bottom,
and sides of a contcmiing vessel. — The total 'pressure due to
gravity on amj portion of the horizontal bottom of a vessel con-
taining a liquid is equal to the iveight of a column of the same
liquid ivhose base is the area of that portion of the botto^n pressed
upon, and whose hight is the depth of the ivater in the vessel.
Thus, suppose that we have three vessels having bottoms of the
same size : one of them has flaring sides, like a wash-basin ;
another has cylindrical sides ; and the third has conical sides,
like a coffee-pot. If the three vessels be filled with water to
the same depth, the total pressure upon the bottom of each
will be equal to the weight of the water in the vessel of
cylindrical shape. Suppose that the area of the bottom of
each is 108 square inches, and the depth of water is 16 inches ;
then the cubical contents of the water in the cylindrical vessel
is 1,728 cubic inches, or 1 cubic foot. The weight of 1 cubic
foot of water is 62.3 pounds. Hence, the total pressure upon
the bottom of each vessel is 62.3 pounds.

Evidently, the lateral pressure at any point of the side of
a vessel depends upon the depth of that point ; and, as depth
at different points of a side varies, to find the total pressure

SURFACE OF A LIQUID AT REST IS LEVEL. 151

upon any portion of a side of a vessel, find the weight of a
column of liquid whose hase is the area of that portion of the
side, and luhose hight is the average depth of that portion,

127. The surface of a liquid at rest is level. -—By jolting a
vessel the surface of a liquid in it may be made to assume tlie
form seen in Eig. 115. Can it retain this form ? Take two
molecules of the liquid at the points a and h on the same level.
The total downward pressures upon a and h are , ^

as their respective depths ca and dh. But l^^^^plj
since (assuming the mass of liquid to be at rest) pi|ip|if||ji|||;li
the pressure at a given depth is equal in all i^nHIPjIf!!
directions, c a and d b represent the lateral pres- 1^— ^~»— ^i
sures at the points a and b res]3ectively. But
db is greater than ca ; hence, the molecules a and h, and those
lying in a straight line between them, are acted upon by two
unequal forces in opposite directions. Hence the liquid can-
not be at rest in the position assumed and there will, therefore,,
be a movement of molecules in the direction of the greater
force, toward a, till there is equilibrium of forces, which will
occur only when the points a and b are equally distant from
the surface ; or, in other words, there ivill be no rest till all
points in the surface are on the same level.

This fact is commonly expressed thus : ' ' Water always seeks its
lowest level." In accordance with this principle, water flows down
an inclined plane, and will not remain heaped up. An illustration
of the application of this principle, on a large scale, is found in the
method of supplying cities with water. Fig. 116 represents a modern
aqueduct, through which water is conveyed from an elevated pond
or river a, beneath a river 6, over a hill c, through a valley cZ, to a
reservoir e, from which water is distributed by service-pipes to the
dwellings in a city. The pipe is tapped at different points, and
fountains would rise to the level of the water in the pond were it
not for the resistance of the air and the check which the ascending
stream receives from the falling drops. Where should the pipes be
made stronger, on a hill or in a valley ? Where will water issue

152

MOLAR DYNAMICS.

from faucets with greater force, in a chamber or in a basement ?
How high may water be drawn from the pipe in the house / ?

128. Artesian ivells, etc. — In most places, the crust of the earth
is composed of distinct layers of earth and rock of various kinds.

Fig. 116.

These layers frequently assume concave shapes, so as to resemble
cups placed one within another. Fig. 117 represents a vertical
section exposing a few of the surface-layers of the earth's crust :
a is a stratum of loose sand or gravel ; &, a clay-bed ; c, a stratum

Fig. 117.

of slate ; d, a stratum of limestone ; the whole resting on a bed of
granite e. If you hollow out a lump of clay, and pour water into
the cavity, you will find that the water will percolate through the
clay very slowly. Water that falls in rain passes readily through
the gravel a, till it reaches the clay-bed 5, where it collects. Hence
a well sunk to the clay-bed, will fill with water as high as the water

ARTESIAN WELLS.

153

stands above the clay. Water also works its way from elevated
places down between the strata of rocks. If a hole be bored
through the slate c, water will rise above the surface of the ground
in a fountain, seeking the level of its source on the hill ; and if bored
still lower, through the stratum d, a still higher fountain may result.
Such borings are called Artesian wells. Water frequently forces
its way through fissures in the rocky strata to the surface, as at i,
and gives rise to springs.

Exercises.

I. The areas of the bottoms of vessels A, B, and C (Fig. 118) are
equal. The vessels have the same depth, and are filled with water.
Which vessel contains the most water ? On the bottom of which vessel
is the pressure equal to the weight of the water which it contains ? How
does the pressure upon
the bottom of vessel
B compare with the
weight of the water
m it?

2. A cubic foot of
water weighs about
62.3 pounds or 1000 ounces. Suppose that the area of the bottom of
each vessel is 100 square inches and the depth is 10 inches ; what is the
pressure on the bottom of each ?

3. Vessel A is a cubical vessel ; what is the total pressure against one
of its vertical sides ?

4. Suppose that vessel A were tightly covered, and that a tube 10 feet
long were passed through a perforation in the cover so that the end
should just touch the upper surface of the water in the vessel ; then sup-
pose the tube to be filled with water. What additional pressure would
each wall of the cube sustain ?

5. Suppose that the area of the end of the large piston of a hydraulic
press is 100 square inches ; what should be the area of the end of the
small piston that a force of 100 pounds applied to it may produce a pres-
sure of 2 tons ?

[Exercises 6 to 10 mclusive should, if practicable, be actually per-
formed in the manner directed, either by the students individually, or by
one or two members of the class m the presence of the rest, while all
work out the results from the data thus obtained.]

154

MOLAR DYNAMICS.

^=^

Fig. 119.

6. Take a glass U-tube (Fig. 119) about 40 in. high, having a stout
rubber tube a attached, and containing mercury witli tlie surfaces at the
same level in both arms. a. Blow into the tube ; the
surfaces of mercury v^ill at once assume different
levels. How will you determine the pressure which
you exert through the air in the tube upon the mer-
cury (the specific density of mercury being 13.59) ?

G7. a. Suck air from a ; what happens to the mer-
cury ? b. How will you determine the diminution of
pressure which you produce by suction ?

8. Take a similar tube containing water instead of
mercury, connect it with a gas jet, and turn on the
gas ; how would you determine how much greater (or less) its pressure
is than that of the atmosphere ?

9. a. Having ascertained as in Exercise 6 the greatest pressure you
can exert by blowmg, how would you proceed to determine the greatest
weight, placed on the piston of the seven-in-one apparatus (Fig. 120), that
you could support, provided there were no friction and the apparatus
were perfectly air-tight ? b. How would you estimate
the loss in force in consequence of friction, etc. ?

10. If the apparatus be inverted and a weight be
hung from the piston, as in Fig. 120, how would you
determine the greatest weight you ought to be able to
raise by suction ?

11. How great is the hydrostatic pressure in fresh
water at the depth 50 feet ?

12. Why does not a person who dives to the bottom
of a pond feel the weight of the column of water above
him ?

13. Why does not the weight of the greater quan-
tity of liquid in a coffee pot when filled cause the
liquid to rise higher in the spout than the surface of
the liquid in the pot ?

14. a. A house is supplied with water by a system of pipes from a
distant reservoir, as is customary in cities ; what data would you require
in order to compute the pressure at any point in the pipe ? b. How
much greater is the pressure at a point in the pipe in the cellar than at
another point in the attic ? c. Is the pressure in the pipe the same when
water is running from a faucet in the house as when the water is at
rest ? Why ?

Fig. 120.

ATMOSPHERIC PRESSURE.

155

15. A (Fig. 121) represents a stand-pipe for furnishing the neighboring
district with water by the action of gravity. The stand-pipe is supplied
with water from a lake in the vicinity by means of a pumping engine.

7n —

Fig. 121.

Vertical distances are represented on a scale of ^ in. =o= 50 ft. If the
stand-pipe be filled to the level mn and the water be at rest in the
main pipe leading from it, what pressure will the pipe sustain at points
a, c, and d respectively ?

Section II.

ATMOSPHERIC PRESSURE.

129. Introduction. — We live at tlie bottom of an exceed-
ingly rare and elastic aerial ocean, called the cdmosphere,
extending to an undetermined distance into space. Every
molecule in the gaseous ocean is drawn towards the earth's
center by gravitation and the atmosphere is thus bound to the
earth by this force, just as is the liquid ocean. Evidently the
pressure in the atmosphere due to its weight increases with
the depth ; or, since in our position we are more accustomed
to speak of hir/ht in the atmosphere, decreases with the hight.
The pressure does not diminish regularly with the hight as
in an ocean of incompressible fluid. Air is very compressible
and therefore varies in density. The lower strata of air
sustaining the weight of air above are relatively much com-
pressed, very dense, and elastic. The density and elasticity
of the air diminish more rapidly than the hight above sea-
level increases. Owing to this fact the greater part of the

156

MOLAR DYNAMICS.

Fig. 122.

mass of the atmosphere is within three and a half miles of
the sea-level (see Fig. 132). Above this hight the air is
much rarefied and vanishes, as it were, very gradually into
empty space.

Experiment 1. — Fill two glass jars (Fig. 122) with water, A having a
glass bottom, B a bottom provided by tying
a piece of sheet-rubber tightly' over the rim.
Invert both in a larger vessel of water, C.
The water in A does \iot feel the downward
pressure of the air directly above it, the
pressure being sustained by the rigid glass
bottom. But it indirectly feels the pressure
of the air on the surface of the water in the
open vessel, and it is this pressure that sustains
the water in the jar. But the rubber bottom
of the jar B yields somewhat to the down-
ward pressure of the air, and is forced inward.
Experiment 2. — Fill a glass tube, D, with water, keeping the lower
end in a vessel of water, and the upper end
tightly closed with a finger. Why does not
the water in the tube fall ? Remove your
finger from the closed end. Why does the
water fall ?

Experiment 3. — Fill (or partly fill) a
tumbler with water, cover the top closely
with a card or writing-paper, hold the
paper in place with the palm of the hand,
and quickly invert the tumbler (Fig. 123).
Why does not the water fall out ?

Experiment 4. — Force the piston A (Fig.
124) of the seven-in-one apparatus quite to
the closed end of the hollow cylinder, and
close the stop-cock B. Try to pull the piston out again. Why do you

not succeed? Hold the apparatus in
various positions, so that the atmosphere
may press down, laterally, and up,
against the piston. Do you discover
any difference in the pressure which it
receives from different directions ?

Fig. 123.

HO\y ATMOSPHERIC PEESSUHE IS MEASURED. 157

130. How atmospheriG pressure is measured.

Experiment 5 (preliminary), —Take a U-sliaped glass tube (Fig. 125),
half fill it with water, close one
end with a thumb, and tilt the
tube so that the water will run
into the closed arm and fill it;
then restore it to its original ver-
tical position. Why does not the
water settle to the same level in
both arms ?

Fig. 125.

-^A

126 represents a U-sliaped glass tube closed at one
end, 34 inches in higlit, and with, a bore of
1 square inch section. The closed arm hav-
ing been filled with mercury, the tube is
placed with its open end upward, as in the
cut. The mercury in the closed arm sinks
about 2 inches to A, and rises 2 inches in
the open arm to C ; but the surface A is 30
inches higher than the surface C. This can
be accounted for only by the atmospheric
pressure. The column of mercury B A, con-
taining 30 cubic inches, is an exact coun-
terpoise for a column of air of the same
diameter extending from C to the upper
limit of the atmospheric ocean, — an un-
known hight.

The weight of the 30 cubic inches of mer-
cury in the column B A is about 14.7 pounds.
Hence the weight of a column of air of 1 square-inch section,
extending from the surface of the sea to the upper limit of
the atmosphere, is about 14.7 pounds. But in fluids gravity
causes equal pressure in all directions. Hence, at the level
of the sea, all bodies are pressed upon in all direetions by the
atmosphere, ivith a force of about 14,7 pounds per square inch,
or about one ton per square foot.

Fig. 126.

158

MOLAR DYNAMICS.

A pressure of 15 pounds per square inch is quite generally
adopted by engineers as a unit of gaseous pressure, and is
called an atmosphere. Physicists, however, generally measure
pressure in terms of cm or mm of mercury at 0° C. ; that is,
the hight in myii of mercury that the pressure of the atmos-
phere sustains in the tube.

Fig. 127.

131. The barometer. — The hight of the col-
umn of mercury supported by atmospheric pres-
sure is quite independent of the area of the
surface of the mercury pressed upon ; hence the
apparatus is more conveniently constructed in
the form represented in Pig. 127.

A straight tube about 34 inches long is closed
at one end and filled with mercury. The tube is
inverted, with its open end tightly covered with
a finger, and this end is inserted into a vessel of
mercury. When the finger is withdrawn the

Fig. 128.

THE FORTIN BAROMETER.

159

mercury sinks until there is equilibrium between the down-
ward pressure of the mercurial column AB and the pressure
of the atmosphere. The empty space at the top of the tube
is called a Torricellian ^ vacuum. An apparatus designed to
measure atmospheric pressure is called a barometer (pressure-
measurer). A common cheaj^ form of barometer is repre-
sented in Pig. 128. Beside the tube and near its top is a
scale graduated in inches or centimeters, indicating the hight
of the mercurial column. For ordinary purposes this scale
needs to have a range of only three or four inches, so as to
include the maximum fluctuations of the column.^

132, The Fortin harometer. — We will suppose the scale of a
barometer to be fixed so as to indicate correctly the hight of the
surface of mercury in the tube above that in the
cistern at a time, for instance, when this distance
is 30 inches. A point on the surface of the mercury
in the cistern in this case is called technically the
zero point. Now should the mercury fall in the
tube to 29 and the mercury in the cistern remain
at zero, then the scale reading would indicate
correctly the barometric hight ; but the mercury
does not remain at zero, but rises a little (less as
the diameter of the cistern is greater) , consequently
the scale-reading is too great. When the mercury
is higher in the tube than 30 all the readings will
be too small. Evidently, then, the mercury in the
cistern must be brought to zero at every obser-
vation in order to eliminate this error. This is
easily accomplished with the Fortin barometer.
The bottom of the cistern of this barometer is
pliable leather resting on a thumb-screw A (Fig.
129). Projecting from the tube inside of the cistern
is a little pointer B of colored glass. The lower end
of this pointer, called the fiducial point, corresponds
to the zero point. The level of the mercury in the
cistern must be set to this point by raising or fig. 129.

1 The first barometer was constructed by Torricelli, a Florentine, in 1643.

2 At the Central Station in Boston, Feb. 8, 1895, the mercury fell to 28.61 in., the
lowest on record at this station. The highest point attained at this station Avas 30.97
in, on Dec. 1, 1887, and Dec. 31, 1889.

=ib

160

MOLAR DYNAMICS.

lowering the cistern base by the adjusting screw, before taking a
reading. A sliding piece C, Fig. 130, furnished with a vernier i
can be slid along the tube so as to enable one to read with great
accuracy.

In refined scientific researches it is necessary to make suitable
allowances for expansion and contraction of the mercury attending
changes in temperature, hence a very sensitive ther-
mometer is attached to the barometer. Also allowances
for capillary depression must be made.

133. The aneroid barometer. — The aneroid (without
moisture) barometer employs no liquid. It contains a
cylindrical box, D (Fig. 131), having a very flexible top.
The air is partially exhausted from within the box. The
varying atmospheric pressure causes this top to rise and
sink much like the chest of a man in breathing. Slight
movements of this kind are communicated by means of
multiplying-apparatus (apparatus by means of which a
small movement of one part causes a large movement of
another part) to the index needle A. The dial is gradu-
ated to correspond with a mercurial barometer. The

observer turns the button C and brings the brass needle B over the

black needle A, and at his next

observation any departure of the

latter from the former will show pre-
cisely the change which has occurred

between the observations.

The aneroid can be made more

sensitive (i.e. so as to show smaller

changes of atmospheric pressure)

than the mercurial barometer.

Owing to this as well as to its con-
venient size and portability, the

aneroid has become quite popular.

Unfortunately, however, it does not

preserve its accuracy for a great

length of time, hence it must be adjusted from time to time to a

standard mercurial barometer.

Fig. 130.

Fig. 131.

1 For construction and methods of using verniers the student is referred to
Pickering's " Manual of Physical Manipulations," or Stewart and Gee's " Elementary
Practical Physics."

STANDARD PRESSURE. 161

134. Standard ])vessiire. — Many physical operations require
a standard pressure for reference. The standard generally
adopted is the pressure exerted by a column of pure mercury
at 0° C. and 76 cm. (29.922 inches) high, which is about the
average hight of the barometric column at sea-level in latitude
45"". The pressure corresponding to this hight is 1033.3
grams per square centimeter or 14.69 pounds per square
inch.

135. Barometric measurement ofhights. — Since atmospheric
pressure varies with the hight above sea-level, it is evident
that changes in elevation may be determined from changes
of pressure as indicated by the barometer. In other words,
the hight of a mountain may be ascertained from barometric
readings made on the summit and at sea-level. Such deter-
minations are more reliable for moderate elevations, as there
are elements of greater or less uncertainty in measuring great
hights. For moderate hights the barometric column falls at
a very nearly uniform rate of one inch for every 900 feet of
ascent.

If a mercurial barometer stand at 760 mm. on the floor, the
same barometer on the top of a table 1 m. high should stand
at a hight of 759.91 mm., a change scarcely perceptible. The
aneroid is, however, sometimes made so sensitive that the
change of pressure experienced in this short distance is ren-
dered quite perceptible.

The shading in Fig. 132 is intended to indicate roughly the
variation in the density of the air at different elevations above
sea-level. The figures in the left margin show the hight in
miles ; those in the first column on the right, the corresponding
average hight of the mercurial column in inches ; and those
in the extreme right, the density of the air compared with its
density at sea-level.

It is calculated that if an opening could be made in the
earth 35 miles in depth below the sea-level, the density of the

162

MOLAR DYNAMICS.

air at the bottom would be 1,000 times that at sea-level, so
that water would float in it.

00 30000

Fig. 132.

If the aerial ocean were of uniform density, and of the same
density that it is at sea-level, its depth would be a little less
than five miles. Only a few peaks of the Himalayas would
rise above it.

136. The barometer in meteorology.'^ — The barometer is some-
times called a "weather-glass," chiefly because its scale frequently
bears the words fair., rainy, storm, etc. These words are very
objectionable, since they are totally wrong from a meteorological
point of view. To form a forecast of the weather of much value, a

1 The following works will he found useful to students of meteorology: "Ele-
mentary Meteorology," by W. M. Davis ; " Instructions in tlie Use of Meteorological
Instruments," " Elementary Meteorology," " Weather Charts and Storm Warnings,"
by H. H. Scott; "Weather Casts and Storm Prevision," by K Strachan; and "A
Treatise on Meteorological Instruments," by Negretti and Zambra.

THE BAROMETER IN METEOROLOGY. 163

barometer, a thermometer, and a hygrometer must be consulted,
and one must be familiar with the laws which govern the relations
between atmospheric pressure, temperature, moisture, etc. In
forming a judgment of forthcoming weather, the point at which the
mercury stands should not be so much regarded as whether it is
rising or falling. The following general rules may not be amiss,
though even these have many exceptions : —

1. A steady barometer at about its mean hight, with a seasonable
temperature, and dry air, indicates a continuation of fine weather.

2. A rise from this point indicates decidedly fine and dry ; a fall
indicates rain or higher wind.

3. A gradual rise or fall indicates a less immediate change than
a more rapid motion of the mercury.

Eluctuations in barometric pressure are of hourly occurrence.
Some of the many conditions which influence the atmospheric pres-
sure are the following -. (1) Temperature. A rise of temperature
tends to diminish the air-pressure. In general the barometer falls
as the thermometer rises. Heat expands the air, causing a lateral
flow away from the heated areas. (2) Humidity. Moist air is
lighter than dry air having an equal pressure. (3) Currents in the
atmospheric ocean. On weather charts, lines called isobars are drawn
through places having the same pressure. Frequently the isobar
indicating the lowest pressure encloses an area more or less circular.
This is called a "center of depression." The surrounding air tends
to flow into it from all sides. The greater the difference of pressure
between the two places, i.e. the steeper the barometric gradient,
the more rapid the flow. The direction of the wind, however, is so
modified by the rotation of the earth that the flow is not directly
toward the center, but spirally toward it, the motion in the northern
hemisphere being opposite to the direction in which the hands of a
watch move.

164

MOLAR DYNAMICS.

Section III.

RELATION BETWEEN THE DENSITY, VOLUME, AND PRESSURE
OF A BODY OF GAS.

137. Boyle's (or Mariotte^s) Law.

Experiment 1. — Take a bent glass tube (Fig. 133), the short arm being
closed, and the long arm, which should be at least 34
inches (85 cm.) long, being open at the top. Pour
mercury into the tube till the surfaces in the two
arms stand at zero. Now the surface in the long
arm supports the pressure of an atmosphere. There-
fore the pressure of the air enclosed in the short arm,
which exactly balances it, must be about 15 pounds
to the square inch. Next pour mercury into the long
arm till the surface in the short arm reaches 5, or till
the volume of air enclosed is reduced one half, when
it will be found that the hight of the column A C is
just equal to the hight of the barometric column at
the time the experiment is performed. It now ap-
pears that the pressure of the air in A B balances the
atmospheric pressure, plus a column of mercury A C
which is equal to another atmosphere ; .-. the pres-
sure of the air in A B = two atmospheres. But the
air has been compressed into half the space it for-
merly occupied, and is, consequently, twice as dense.
If the length and strength of the tube would admit of
a column of mercury above the surface in the short
arm equal to twice A C, the air would be compressed
into one third its original bulk ; and, inasmuch as it
would balance a pressure of three atmospheres, its
pressure would be increased threefold.

This experiment may be conducted in a more sci-
entific manner, as follows : —

Experiment 2, — Let the mercury be at the same

level, AB (Fig. 134), in both arms of the tube. The

body of air to be experimented with is in the short arm between A and

C. The dimensions of this body can vary only in hight ; hence its

hight, H, may represent its volume. Measure H, i.e. the distance

Fig. 133.

BOYLE S LAW.

165

between A and C, and regard the number of inches (or centimeters)
as representing the volume, V. Its pressure, P, evidently is the same as
that of the atmosphere at the time of experimenting.
Consult a barometer, and ascertain the hight of the baro-
metric column ; let this hight represent P. Pour a little
mercury into the tube ; the mercury rises to Ai and Bi.
Measure from Ai to C ; this number represents the vol-
ume, Fi, of the body of gas now. Measure the vertical
distance between Ai and Bi ; this number represents the
increase in pressure, which, added to P, will give its
present pressure, Pi. [— Bs

Now pour more mercury into the long arm, so that it
win rise to some such points as A2 and Bg. Determine
as before the new volume, Fo, and new pressure, P2.
So continue to add mercury a third, and a fourth time,
and get new values for the volume, F3, and F4, and
for the pressure, P3, and P4. Arrange the results as
follows : —

F =
Fi-

etc.

P =

Pi =

P2 =

etc.

V X P =
Vi xp,=

etc.

A2-

A-

-B

Fig. 134.

It will be found that the series of products in the last
column are approximately equal (due allowance being made
for errors in measurement, etc.) ; consequently F varies
inversely as V. Hence the law : —

The volume of a body of gas at a constant temperature varies
inversely as its pressure, density, and elasticity.

For many years after the announcement of this law, first
by Boyle and a little later by Mariotte, it was believed to
be rigorously correct for all gases ; but more recently, more
precise experiments have shown that it is approximately but
not rigidly true for any gas, that the departure from the law
differs with different gases, and that each gas possesses a
special law of compjressibility. There is a limit beyond which
this law does not hold. This limit is soonest reached with
those gases, like carbon-dioxide, chlorine, etc., that are most

166

MOLAR DYNAMICS.

ary

readily liquefied by pressure. A gas is nearer perfect, or
conforms more nearly to Boyle's law, in proportion as it is at
a greater distance, as regards both temperature and pressure,
from its liquefying point. When a gas is near the critical
state its density increases more rapidly than its elasticity.^

138. Manometer, or pressure gauge. — The manometer, an instru-
ment for measuring the pressure of a gas or vapor contained in a
closed vessel, under considerable pressure, illustrates the application
of Boyle's law to a practical purpose.

This instrument consists, as in Fig. 135, of a bent tube A B closed
at one end a, and containing within the space
Aa a quantity of air, which is cut off from
external communication by a column of mer-
cury. The apparatus is so constructed that
when the pressure on B is equal to that of an
atmosphere, the mercury stands at the same
hight in both branches. But if the pressure
increase, the mercury is forced into the left
branch, so that the air in that branch is com-
pressed, and its elasticity proportionately in-
creased. The pressure of the gas exerted at B
is then equal to the pressure of the compressed
air, together with that of a column of mercury
m n equal to the difference of level of the liquid in the two branches.
This pressure is expressed in atmospheres on the scale ab.

139. Elasticity of gases. — The elasticity of all fluids is
perfect. By this is meant, that the force exerted
in expansion is, except within certain limits
rarely reached, equal to the force used in com-
pression; and that, however much a fluid is
compressed, it will always completely regain
its former bulk when the pressure is removed.

Hence the barometer, which measures the com- ^L -t^

pressing force of the atmosphere, also measures ^^^^"^^T^^"^

at the same time the elasticity of the air.

A so-called vacuum gauge (Fig. 136) is simply a short mer-

1 The student will find this subject admirably treated in Daniell, pp. 204, 222, etc.

Fig. 135.

SUCTION.

167

cury barometer, — short because it is seldom required to
make measurements except in tolerably high vacua, where
the mercurial column is correspondingly low. For instance,
this apparatus, placed under the receiver of an air-pump
from which air is exhausted, will measure the elasticity of
the air in the receiver. This known, the degree of exhaustion
is readily determined.

Experiment 3. — Force the piston of the seven-in-one apparatus two
thirds of the way into the cylinder, and close the aper-
ture. Support the apparatus on blocks, with the piston
upwards, and place the whole under the receiver of an
air-pump. Exhaust the air from the receiver ; the outside
pressure of the air being partially removed, the unbalanced
pressure of the air enclosed within the cylinder will cause
the piston to rise.

Experiment 4. — Take a glass tube (Fig. 137) having a
bulb blown at one end. Nearly fill it with water, so
that when it is inverted there will be only a bubble of air
in the bulb. Insert the open end in a glass of water, place under a
receiver, and exhaust. Nearly all the water will leave the bulb and tube.
Why ? What will happen when air is admitted to the
receiver ?

140. Suction. — Liquids are said to be raised
by the "force of suction," which seems to imply
that a lifting force acting from above ^:>?6Z^s- the
liquid up. Does this explain suction ?

Fig. 137,

Experiment 5. — Fill a glass U-tube having unequal
arms (Fig. 138) with water to the level c h. Close the end
6 with a finger, and try to suck the liquid out of the tube. You find it
impossible. Eemove the finger from 6, and you can suck the liquid out
with ease. Why ?

168

MOLAR DYNAMICS.

Sectiox ly.

INSTRUMENTS USED FOE, RAREFYING AIR.

141. The aiT-puin2J. — The air-pump

FlCx. 139.

is used to rarefy air
in a closed vessel.
Fig. 139 will serve
to illustrate its
operation. E is a
glass receiver within
which the air is to
be rarefied; B is a
hollow cylinder of
brass, called the
picmp -barrel ; the
plug P, called a ins-
ton, is fitted to the
interior of the bar-
rel, and can be moved up and down by the handle H ; *' and t
are valves. A valve acts on the principle of a door intended
to open or close a passage. If you walk against a door on
one side, it opens and allows you to pass ; but if you walk
against it on the other side, it closes the passage, and stops
your progress. Suppose the piston to be in the act of
descending ; the compression of the air in B closes the valve
t, and opens the valve s, and the enclosed air escapes. After
the piston reaches the bottom of the barrel, it begins its
ascent. This would cause a vacuum between the bottom of
the barrel and the ascending piston (since the unbalanced
pressure of the outside air immediately closes the valve s),
but the pressure of the air in the receiver E, opens the valve t
and fills this space. As the air in E expands, it becomes
rarefied and exerts less pressure. The external pressure of
the air on E, being no longer balanced by the pressure of

INSTRUMENTS USED FOR RAREFYING AIR.

169

the air within, presses the receiver firmly upon the plate L.
Each repetition of a double stroke of the jjiston removes a
portion of the air remaining in R. The air is removed from
E by its own expansion. However far the process of exhaus-
tion may be carried, the receiver will always be filled with
air, although it may be exceedingly rarefied. The operation
of exhaustion is practically ended when the pressure of the
air in R becomes too feeble to lift the valve t, unless the
apparatus be so constructed that the valves are opened and
closed by mechanical action. It is obvious that if s and t
opened downward instead of upward, then as the piston is
raised and depressed, air would be compressed in R. A con-
denser is merely a pump with its valves reversed, and is used
to condense air.

142. The mercury air-pump. — In recent years the so-called
mercury air-pump has largely displaced the pump described
above, since it is capable of producing a much greater rare-
faction. In brief, it makes use of the Torricellian vacuum,
such as is formed in the top of a barometer tube. On account
of its simplicity, the Geissler pump, the first of the kind
invented, is chosen for illustration. A (Fig. 140) is a glass
tube more than thirty- e

four inches long, having ^ ^M^^

a globe-like enlargement
B of about a liter capac-
ity. Above this globe
leads a tube containing a
three-way stop-cock and
a branch tube D. By
means of this stop-cock
B may be placed in com-
munication either through
D with the atmosphere,
as shown in M (Fig. 141), or through E with the receiver to

Fig. 140.

B
M
Fig. 141.

170 MOLAR DYNAMICS,

be exhausted, as shown in N. Connected with, the lower end
of A by means of a thick rubber tube is a vessel G containing
mercury. The pump is operated as follows : C is turned as
in M, G- is raised so that mercury will flow from it into B
and fill it, the air escaping through D. Then the stop-cock is
placed as in 'N, and G- is lowered so as to allow the mercury
to flow back into it. A Torricellian vacuum would be formed
in B were it not in communication through E with the
receiver. As it is, the air in this space expands and fills B,
and is thus to this extent rarefied. By a sufficient number
of repetitions of this process, a very high vacuum is obtaina-
ble. There are many modifications of this pump, in some of
which the stop-cock is dispensed with, and consequently the
trouble of operating it is avoided.

With the common pump a vacuum of a millimeter of mer-
cury is considered exceedingly good ; but with a mercury pump
it is easy to obtain a vacuum of .00076 of a millimeter, which
represents about one millionth the normal pressure of the
atmosphere. "Eood in 1881 succeeded in obtaining vacua
as high as a three hundred-millionth of an atmosphere." —
Barke7\

Section V.

SIPHONS AND PUMPS FOR LIQUIDS.

143. Construction and operation of the siphon. — A siphon
is an instrument used for transferring a liquid from one vessel
to another over an elevation through the agency of atmo-
spheric pressure. It consists of a tube of any material (rub-
ber is often most convenient) bent into a shape somewhat
like the letter U. To set it in operation, fill the tube with
a liquid, stop each end with a finger or cork, place it in the
position represented in Fig. 142, remove the stoppers and
the liquid will flow out at the orifice o. Why? The up-

SIPHONS AND PUMPS FOR LIQUIDS. 171

ward pressure of the atmosphere against the liquid in the
tube is the same at both ends ; hence these two forces are
in equilibrium. But the downward

pressure of the column of liquid ah T ~Jr\ "i"

is greater than the downward pres- ^' ^^P \\~ ^^

sure of the column d c ; hence equi- ^^S| \\ j

librium is destroyed and the move- \\ i

\\ j

ment is in the direction of the &\—ih

greater (i.e. the unbalanced) force.

The unbalanced force which causes the flow is equal to the
downward pressure of the column e h.

If one end of the tube filled with liquid be immersed in a
liquid in some vessel, as in A (Fig. 143), and the other end
be brought below the surface of the liquid in the vessel and
the stoppers be removed, the liquid in the vessel will flow out
through the tube as long as the distance e h remains greater
than zero.

If one of the vessels be raised a little, as in C, the liquid will flow
from the raised vessel, till the surfaces in the two vessels are on the
same level. The remaining diagrams in this cut represent some
of the great variety of uses to which the siphon may be put. D, E,
and F are different forms of siphon fountains. In D, the siphon
tube is filled by blowing in the tube /. Explain the remainder of
the operation. A siphon of the form G is always ready for use. It
is only necessary to dip one end into the liquid to be transferred.
Why does the liquid not flow out of this tube in its present con-
dition ? H illustrates the method by which a heavy liquid may be
removed from beneath a lighter liquid. By means of a siphon a
liquid may be removed from a vessel in a clear state, without
disturbing sediment at the bottom. I is a Tantalus cup. A liquid
will not flow from this cup till the top of the bend of the tube is
covered. It will then continue to flow as long as the end of the
tube is in the liquid. The siphon J may be filled with a liquid that
is not safe or pleasant to handle, by placing the end j in the liquid,
stopping the end jt, and sucking the air out at the end I till the
lower end is filled with the liquid.

172

MOLAR DYNAMICS.

Gases heavier than air may be siphoned like liquids. Vessel o
contains carbonic acid gas. As the gas is siphoned into the vessel p,
it extinguishes a candle-flame. Gases lighter than air are siphoned
by inverting both the vessels and the siphon.

Fig. 143.

The siphon cannot elevate liquids, it can merely transfer
liquids to places of lower level. It is apparent that the
pressure of the air drives the liquid through the siphon, and
that a siphon would be inoperative in a vacuum. Obviously
the hight of the bend to which different liquids can be raised
varies with their respective densities. Thus the hight to
which mercury can be raised is the barometric hight, while
water may be raised 13.6 times as high.

SIPHONS AND PUMPS FOR LIQUIDS.

173

144. Lifting or suction puinp. — The common lifti7ig-pump
is constructed like the barrel of an air-pump. Fig. 144 rep-
resents the piston B in the act of rising. As the air is rarefied
below it, water rises in consequence of atmospheric pressure
on the water in the well, and opens the lower valve D.

Fig. 145.

Fig. 146.

Fig. 144.

Atmospheric pressure closes the upper
valve C in the piston. When the piston is
pressed down (Fig. 145), the lower valve
closes, the upper valve opens, and the
water between the bottom of the barrel
and the piston passes through the upper
valve above the piston. When the piston is raised again
(Fig. 146), the water above the piston is raised and discharged
from the spout.

The liquid is sometimes said to be raised in a lifting-pump
by the "force of suction." Is there such deforce?

Calling the specific density of mercury 13.6 and disregarding
the vapor pressure of mercury (.02 mm at 20° C), a water
barometer would be 76 X 13.6 = 1033.6 cm high when the
mercury barometer stands at 76 cm, provided there were no
pressure exerted by the vapor of water. But at 20° C. the

1T4

MOLAR DYNAMICS.

pressure of water vapor is 1.74 cm of mercury = 1.74 X 13.6
(=^ 23.7 cm) centimeters of water. Therefore the hight of the
water barometer would be 1033.6 — 23.7 = 1009.9 cm, and this
is the limit of the hight to which water can be raised by the
pressure of air in a suction pump under these conditions.^

145. Suction and foixe-pumjj comhined. — In this pump the
ordinary piston with valve and leather washers is replaced
by a solid cylinder of metal, B (Fig. 147), called the plunger.

This passes through a stuffing box
D, in which it fits air-tight. Valves
opening upward and outward are
placed at A and C respectively.
When the plunger is raised, A
opens and C closes, and water is
raised into the barrel by atmos-
pheric pressure. When the plunger
descends, A closes and C opens, and
the water is forced up through the
pipe E to a hight dependent on the
pressure brought to bear upon it
through the plunger. An air-dome
F is usually connected with these
pumps to regulate the pressure so
as to give through the delivery pipe a very steady stream.
This dome contains air. When the plunger descends it forces
water violently into E and thus tends to produce a severe
strain in the pipe. But the water enters the dome, compresses
the elastic air within, and thus the shock is largely reduced.
As soon as the down stroke of the piston ceases, the valve C
closes, and the compressed air in the dome forces the water
out through E in a continuous stream.

Fig. 147.

1 At 50° the limit would be 1033.6 - (9.2 x 13.6) = 908.4 cm. At 100° the hight would
be 1033.6 — (76 x 13.6) = cm, the pressure of water vapor being equal to the pressure
of the atmosphere.

BUOYANCY OF FLUIDS. 175

Section VI.

BUOYANCY OF FLUIDS.

146. Whi/ a solid is huoijecl up by a fluid, and ivith hoiv
great a force it is buoyed up. — Suppose dob a (Fig. 148) to
be a cubical block of marble immersed in a liquid. It is
obvious tliat the downward pressure upon the surface da is
equal to the weight of the column of liquid
edao. The upward j)i'essure on the surface '^ ^ ^

c^ is equal to the weight of a column of
liquid ecbo. The difference between the
upward pressure against cb and the down-
ward pressure on da, is the weight of a
column of liquid ecbo leslled with mercury the block weighs 570 grams. What
is the capacity of the cavity ?

26. In which is it easier for a person to float, in fresh water or in sea-
water ? Why ?

27. Eig. 156 represents a beaker graduated in
cubic centimeters. Suppose that when water stands
in the graduate at 50 cc, a pebble stone is dropped
into the water, and the water rises to 75 cc. a.
What is the volume of the stone ? 5. How much
less does the stone weigh in water than in air ? c.
What is the weight of an equal volume of water ?

28. If a piece of cork be floated on water in a
graduate, and displace {i.e. cause the water to rise)
7 cc, what is the weight of the cork ?

29. You wish to measure out 50 g of sulphuric
acid. To what number on a beaker graduated in
cubic centimeters will that correspond ?

30. State how you would measure out 80 g of nitric acid in a meas-
uring-beaker.

Fig. 156.

184 MOLAR DYNAMICS.

31. A measuring-beaker contains 35 cc of naphtha. What is the
weight of the naphtha ?

32. If 15 g of salt be dissolved in 1 liter of water without increasing the
volume of the liquid, what will be the specific density of the solution ?

33. A mass of lead weighs 1 K in air. What will it weigh in a
vacuum ?

34. A mass whose weight in air is 30 g, weighs in water 26 g, and
in another liquid 27 g. What is the specific density of the other
liquid ?

35. A silver spoon weighing 150 g is supported by a string in water.
What part of the weight is sustained by the string, and what part is sup-
ported by the water ?

36. Find the specific density of wax from the following data : weight
of a given mass of wax in air is 80 g ; wax and sinker displace 102.88 cc
of water ; sinker alone displaces 14 cc.

37. A boat displaces 25 m^ of water. How much does it weigh ?

38. If 50 K of stone were placed in the boat, how much water would
it displace ?

39. If the boat be capable of displacing 100 m^ of water, what weight
must be placed in it to sink it ?

40. An empty glass globe weighs 100 g ; full of air it weighs 102.4 g ;
full of chlorine gas, it weighs 105.928 g. What is the specific density of
chlorine gas ?

41. What mass of alcohol can be put into a vessel whose capacity is
1 liter ?

42. A solid floats at a certain depth in a liquid when the vessel which
contains it is in the air ; if the vessel be placed in a vacuum, will the
solid sink, rise, or remain stationary ?

43. When the volume of a body of gas diminishes, is it due to con-
traction or compression, i.e. to internal or external forces?

44. What is the hight of the barometer column when the atmospheric
pressure is 10 grams per square centimeter ?

45. A barometer in a diving-bell stands at 196 cm when a barometer
at sea-level stands at 76 cm ; what is the depth of the surface of water
inside the bell, below the air-exposed surface of the water above ?

46. A measuring glass graduated in cubic centimeters contains water.
An empty bottle floats on the water, and the surface of the water stands
at 50 cc. If 10 g of lead shot be placed in the bottle, where will the
surface of the water stand ?

47. A person can lift just 200 K of copper in water ; how much can
he lift in the air ?

EXERCISES. 185

48. If a liter of gas under a pressure of 76 mm be allowed to expand
and fill a vessel having a capacity of 10 liters, v^hat pressure will it
exert ?

49. A piece of lead and a piece of cork balance each other in the air.
Which contains more matter, and how much more ?

50. How great a buoyant force does a fluid exert on a body immersed
in it?

186 MOLAR DYNAMICS.

CHAPTEE V.

ENERGY OF MASS VIBRATION. — SOUND-WAVES.

Section I.

ORIGIN OF SOUND-WAVES ; TRANSMISSION OF SOUND-WAVES.

152. How sound originates. — Listen to a sounding church
bell. It produces a sensation ; it is heard. The ear is the
organ through which the sensation of hearing is produced.
The bell is at such a distance that it cannot act directly on
the ear ; yet something must act on the ear, and it must be
the bell which causes that something to act.

How does a sounding body differ from a silent body ?

Experiment 1. — Strike a bell or a glass bell-jar, and touch the edge
with a small ivory ball suspended by a thread ; you not only hear the
sound, but, at the same time, you see a tremulous motion of the ball,
caused by a motion of the bell. Touch the bell gently with a finger, and
you feel a tremulous motion. Press the hand against the bell ; you stop
its vibratory motion, and at that instant the sound ceases. Strike the
prongs of a tuning-fork, and press the stem against a table ; you hear a
sound. Thrust the ends of the prongs just beneath the surface of water ;
the water is thrown off in a fine spray on either side of the vibrating fork.
Watch the strings of a piano, guitar, or violin, or the tongue of a jews-
harp, when sounding. You can see that they are in motion.

Sound originates in niass-vibratio7i.

How can a bell sounding at a distance affect the ear ? If
the bell while sounding possess no peculiar property except
motion, then it has nothing to communicate to the ear but
motion. But motion can be communicated by one body to
another at a distance only through some medium.

HOW SOUND ORIGINATES. 187

Experiment 2. — Lay a thick tuft of cotton-wool on the plate of an
air-pump, and on this, face downward, place a loud-ticking watch, and
cover with the receiver. Notice that the receiver, interposed between
the watch and your ear, greatly diminishes the sound, or interferes with
the passage of something to the ear. Take a few strokes of the pump
and listen ; the sound is more feeble, and continues to grow less and less
distinct as the exhaustion progresses, until either no sound can be heard
when the ear is placed close to the receiver, or an extremely faint one,
as if coming from a great distance. The rehioval of air from a portion of
the space between the watch and your ear destroys the sound. Let in
the air again, and the sound is restored.

The vibrations of a sonorous body cannot affect the organ of
hearing without a continuous medium of communication between
them.

153. Hoiu vibratory motion, i.e. a luave, is propagated through
an elastic medium.

Experiment 3. — Fig. 157 represents a brass wire wound into the form
of a spiral spring, about 12 feet long. Attach one end to a cigar-box,
and fasten the box to a table. Hold the other end of the spiral firmly in
one hand, and with the other hand insert a knife-blade between the turns
of the wire, and quickly rake it for a short distance along the spiral
toward the box, thereby crowding closer together for a little distance (B)

Fig. 157.

the turns of wire in front of the hand, and leaving the turns behind
pulled wider apart (A) for about an equal distance. The crowded part
of the spiral may be called a condensation^ and the stretched part a rare-
faction. The condensation, followed by the rarefaction, runs with great
velocity through the spiral, strikes the box, producing a sharp thump ; is
reflected from the box to the hand, and from the hand again to the box,
producing a second thump ; and by skillful manipulation three or four
thumps will be produced in rapid succession. If a piece of twine be tied
to some turn of the wire, it will be seen, as each wave passes it, to receive
a slight jerking movement forward and backward in the direction of the
length of the spiral.

188 MOLAR DYNAMICS.

How is energy transmitted through the spring so as to
deliver the blow on the box? Certainly not by a bodily
movement of the spiral as a whole, as might be the case if it
were a rigid rod. The movement of the twine shows that the
only motion which the coil undergoes is a vibratory movement
of its turns. Here, as in the case of water-waves, energy
is transmitted through a medium by the transmission of
vibrations.

The effect of applying force with the hand to the spiral
spring is to produce in a certain section, B, of the spiral a
crowding together of the turns of wire, and at A a separation ;
but the elasticity of the spiral instantly causes B to expand,
the effect of which is to produce a crowding together of the
turns of wire in front of it, in the section C, and thus a for-
ward movement of the condensation is made. At the same
time, the expansion of B causes a filling up of the rarefaction
at A, so that this section is restored to its normal state. This
is not all : the folds in the section B do not stop in their
swing when they have recovered their original position, but,
like a pendulum, swing beyond the position of rest, thus
producing a rarefaction at B, where immediately before there
was a condensation. Thus a forward movement of the rare-
faction is made, and thus a pulse or wave is transmitted with
uniform velocity through a spiral spring or any elastic
medium.

A wave cannot be transmitted through an inelastic soft
iron spiral. Elasticity is essential in a medium, that it may
transmit waves covfijposed of condensations and rarefactions;
and the greater the elasticity, the greater the facility and
rapidity with which a medium transmits waves.

154. Air as a medium of luave-motion.

Experiment 4. — Place a candle flame at the orifice a of the long tin
tube A (Fig. 158) and strike the table a sharp blow with a book near the

AIR AS A MEDIUM OF WAVE-MOTION.

189

orifice b. Instantly the candle flame is quenched. The body of air in
the tube serves as a medium for transmission of motion to the candle.

Fig. 158.

Is the motion transferred that of a current of air through the tube (a
miniature wind), or is it a vibratory motion ? Burn touch-paper i at the
orifice 6, so as to fill this end of the tube with smoke, and repeat the last
experiment.

Evidently, if the body of the air be moved along through
the tube, the smoke will be carried along with it. The candle
is blown out as before, but no smoke issues from the orifice a.
It is clear that there is no translation of material particles
from one end to the other, — nothing like the flight of a rifle
bullet. The candle flame is struck by something like a ^^ulse
of air, not by a luind}

Air is a fluid, and has therefore only volume elasticity.
The only waves it can propagate are waves composed of com-
pressions and rarefactions. In a previous chapter we have
seen how a wave is the result of a transmission of harmonic
motion or harmonic vibrations through a series of particles. A
sound-wave consists of a succession of particles of the sound
medium vibrating harmonically and successively and in the

1 To prepare touch-paper, dissolve about a teaspoonful of saltpetre in a half -tea-
cupful of hot water, dip unsized paper in the solution, and then allow it to dry. The
paper produces much smoke in burning, but no flame.

2 If a membrane be tied tightly over the orifice h and a sudden blow be given it
{e.g. by snapping it with a finger), the vibratory character of the motion communi-
cated through the tube is well shown by the flame being first driven from the orifice a
and immediately afterward drawn toward it.

190 MOLAR DYNAMICS.

same direction as that in which the sound-wave moves. There
are two important distinctions between these waves and waves
of water, or waves sent along a cord when one end is shaken :
the former consist of condensations and rarefactions ; the
latter of elevations and depressions. In the former, the
vibration of the particles is "in the same line Avith the path of
the wave, and hence they are called longitudinal vibrations ;
in the latter the vibrations take place in planes at some angle
to the path of the wave, and are therefore called transverse
vibrations.

Boys often amuse themselves by inflating paper bags, and
with a quick blow bursting them, producing with each a single
loud report. First the air is suddenly and greatly condensed
by the blow, and the bag is burst ; the air now, as suddenly
and with equal force, -expands, and by its expansion condenses
the air for a certain distance all around it, leaving a rare-
faction where just before had been a condensation. If many
bags were burst at the same spot in rapid succession, the
result would be that alternating shells of condensation and
rarefaction would be thrown off, all having a common center,
enlarging as they advance, as do the ripples formed by stones
dropped into water ; except that, in this case, the waves are
not like rings, but hollow globes ; not circular, but spherical.
In this manner sound-waves produced by the vibration of a
sounding body travel through the air.

As a wave advances, each individual air-particle concerned
in its transmission performs a short excursion to and fro in
the direction of a straight line radiating from the center of
the shells or hollow globes. A sound-ivave travels its oivn
length in the time that a jparticle occupies in going through one
complete vibration so as to he ready to start again.

Experiment 5. — Take a strip of black cardboard 4.5 inches X 1 inch.
Cut a slit about one-sixteenth of an inch wide lengthwise and centrally
through the strip nearly from end to end. Place the slit across the page

NATURE OF SOUND AND SOUND-WAVES.

191

just below Fig. 159, and draw the book along underneath in the dh-ection
of the arrow. Imagine that the short dark dashes seen through the slit
represent a series of air-particles, and the slit itself represents the direc-
tion in which a series of sound-waves is travelling. It will be seen that

Fig. 159.

each air-particle moves a little to and fro in the direction m which the
sound-waves travel, and comes back to its starting-point ; but the conden-
sations and rarefactions, represented by a group (half a wave-length) of
dots becoming alternately closer together or farther apart, are transmitted
through the whole series of air-particles.

155. Nature of sound and sound-ivaves. — Sound is a sen-
sation caused usually hy air-waves heating upoii the organ of
hearing}

1 As commonly used tlie term sound is ambiguous, being applied to both a
sensation and the physical cause of the sensation. In a scientific treatise ambiguity
and consequent confusion are disastrous. No apology, therefore, is required for
restricting the term to its legitimate signification. With sound itself^'e have little

192 MOLAR DYNAMICS.

Sound-waves are waves in any mediuni (usually air) that are
capable of producing the sensation of sound. A body vibrating
in an elastic medium, e.g. in air, does not necessarily produce
sound-waves ; in other words not all waves are sound-waves.
For example, the energy of the vibrations may be insufficient,
or the vibrating body may be so small (or the medium so
rare) that it cuts through the medium without condensing it
sufficiently to produce audible effects.

156. Solids and liquids are media capable of transmitting
sound-waves.

Experiment 6. — Lay a watch, with its back downward, on a long
hoard (or table), near to one of its ends, and cover the watch with loose
folds of cloth until its ticking cannot be heard through the air in any
direction at a distance equal to the length of the board. Now place the
ear in contact with the farther end of the board, and you will hear the
ticking of the watch very distinctly.

Experiment 7. — Place one end of a long pole on a cigar box, and
apply the stem of a vibrating tuning-fork to the other end; the sound-
vibrations will be transmitted through the pole to the box, and a sound
will be given out by the box, as though that, and not the tuning-fork,
were the origin of the sound.

Experiment 8. — Place the ear to the earth, and listen to the rumbling
of a distant carriage ; or put the ear to one end of a long stick of timber,
and let some one gently scratch the other end with a pin.

Section II.

SPEED OF SOUND-WAVES.

157. Speed of sound-waves dependent on elasticity and den-
sity of mediuin. — It may be demonstrated -^ that in simple
harmonic motion, the velocity with which a particle of an elastic

to do, as this is a physiological rather than a physical phenomenon. No more
appropriate name than sound-wave can be applied to the physical agent with which
we are to deal ; it stiggests at once the reality, and is not suggestive of some vague
mysterious thing shot through space.
1 See Barker's Physics, p. 219.

SPEED OF SOUND-WAVES. 193

medium vibrates, and therefore the speed of propagation in the
TYiedium (Jience, the speed of a sound-wave^, is directly propor-
tional to the square root of the elasticity of the medium, and
inversely proportional to the square root of its density. The
relation of these quantities is shown in the formula

If the elasticity and density of the medium vary alike, and
in the same direction, it is evident that the speed of the
sound-wave is unaffected. Hence the speed of a sound-wave
is unaffected by barometric hight, or elevation above sea-level.
Temperature, however, affects only the density of air. Ele-
vation of temperature of the air diminishes the density of the
air, and therefore tends to increase the speed of the sound-
wave. Moisture in the air renders it less dense (pressure
remaining constant), and thereby tends to increase the speed.
The velocity of a sound-wave is greatest in the direction of
the wind. Speed of sound-waves is very nearly independent
of pitch and intensity.

The greater density of solids and liquids, as compared with
gases, tends, of course, to diminish the speed of sound-waves;
but their greater incompressibility more than compensates
for the decrease of speed occasioned by the increase of den-
sity. As a general rule, solids are more incompressible than
liquids ; hence sound-waves generally travel faster in the
former than in the latter. For example, sound-waves travel
in water about four times as fast as in air, and in iron and
glass sixteen times as fast.

The speed of sound-waves in free air at 0° C. is 332.4 m
(nearly 1091 ft.) i^er second. The increase of speed per
degree C. is .608 m (23.9 in.). The speed in other gases =

—j^, in which v is the speed in air and d the density of the
given gas referred to air. For example, in hydrogen, whose

194 MOLAR DYNAMICS.

density is ^ that of air, the speed at 0° C. is about 4163 ft.
per second. The speed of sound-waves in any medium may
be calculated from the formula given above and by experi-
mental methods to be given further on.

Section III.

ENERGY OF SOUND-WAVES. LOUDNESS. '

158. Energy of sound-ivaves depe7ids on the amplitude of
vibration. — Gently tap the prongs of a tuning-fork and dip
them into water, — the water is scarcely moved by them ;
increase the energy of the blow, — the vibrations become
wider, and the water spray is thrown with greater force and
to a greater distance. The same thing occurs when the fork
vibrates in the air ; though we do not see the air-particles as
they are batted by the moving fork, yet we feel the effects
as a sound sensation, and we judge of their energy by the
intensity of the sensation which they produce.

Fix your attention upon a particle of air as a sound-wave
passes it. A harmonic motion is. impressed upon it. At a
certain point of its excursion its velocity is at its maximum.
Now since the energy of a moving particle varies as the
square of its velocity, the intensity of the impact which it is
capable of producing upon the tympanum of the ear is propor-
tional to the square of this maximum velocity.

It is also clear that if the amplitude of vibration of a
particle be doubled while its period remains constant, its
velocity is doubled, and therefore its energy is increased four-
fold. Hence, (1) measured mechanically, the energy of a
sound-wave is proportional to the square of the amplitude of the
vibration of particles, or, it is proportional to the square of the
maximum velocity of the vibrating particles. An amplitude of
less than to-q oV"o o o ^"^ ^^ sufficient to cause hearing.

ENERGY OF SOUND-WAVES. 195

Loudness of sound refers to the intensity of a sensation.
We have no standard of measurement for a sensation, so we
are compelled to measure the energy of the sound-wave,
knowing at the same time that loudness is not projjortional to
this energy.

159. Energy of sound-ivaves depends iipjon the density of
the niedmm. — In the experiment with the watch under the
receiver of the air-pump (p. 187), the sound grew feebler as
the air became rarer. Aeronauts are obliged to exert them-
selves more to make their conversation heard when they
reach great hights than when in the denser lower air. In
diving-bells persons are obliged to speak in undertones. In a
rare medium a vibrating body during a single vibration either
sets in motion fewer particles, as in the case of the partially
exhausted receiver, or, as in the case of hydrogen gas, it sets
in motion particles of less mass than in a dense medium ;
consequently it parts with its energy more slowly, and the
sound is consequently weaker.

(2) The energy of gaseous sound-waves increases with the
density of the niediimi in which they are produced.

160. Energy of sound-waves depends on distance from their
source. — It is a matter of every-day observation that the
loudness of a sound diminishes very rapidly as the distance
from the source of the waves to the ear increases. As a
sound-wave advances in an ever-widening sphere, a given
quantity of energy becomes distributed over an ever-increasing
surface ; and as a greater number of particles partake of the
motion, the individual particles receive proportionately less
energy ; hence it follows, — as a consequence of the geomet-
rical truth, that " the surface of a sphere varies as the square
of its radius," — that (3) the energy of a sound-wave varies
inversely as the square of the distance from the source. This is
known as the Law of Inverse Squares.^ For example, if two

1 That the Law of Inverse Squares is applicable to sound-waves is sometimes

196 MOLAR DYNAMICS.

persons, A and B, be respectively 500 and 1000 rods from a
gun when it is discharged, the waves that reach A will have
four times the energy that the same waves have when they
reach B.

161. Speaking-tubes.

Experiment 9. — Place a watch at one end of the long tin tube (Fig.
158), and the ear at the other end. The ticking sounds very loud, as
though the watch were close to the ear.

Long tin tubes, called speaking-tubes, passing through many
apartments in a building, enable persons at the distant ex-
tremities to carry on conversation in a low tone of voice,
while persons in the various rooms through which the tube
passes hear little or nothing. The reason is that the sound-
waves which enter the tube are prevented from expanding,
consequently the energy of the sound-waves is not affected by
distance, except as it is wasted by friction of the air against
the sides of the tube, and by internal friction due to the
viscosity of the air.

162. Energy of soimd-iuaves depends on the homogeneousness
of the transmitting medium. — Observations and experiments of
Humboldt, Tyndall, and Henry have established the following
facts : Eain, hail, snow, and fog offer little or no obstruction
to the passage of a sound-wave. The air associated with a
fog is, as a general rule, highly homogeneous and favorable
to the transmission of sound. An atmosphere optically opaque
may be acoustically transparent, and vice versa; hence the
great value of fog horns. Streams of air differently heated,
or saturated in different degrees with aqueous vapor, though
invisible to the eye, form acoustic clouds which may greatly
interfere with the propagation of sound-waves.

163. Energy of sound-ivaves affected by winds. — It not
infrequently happens that sound-waves are audible two or

called in question. For an experimental demonstration of its applicability, see
" Contributions from the Physical Laboratory, Mass. Inst. Technology, 1876."

REFLECTION OF SOUND-WAVES.

197

three times as far to the leeward as to the windward. Sound-
waves are borne along with the wind, but, of course, are
impeded by it when the directions of their motions are opposed
to it.

Section IV.

CHANGES IN DIRECTION OF PROPAGATION OF SOUND-WAVES.

164. Beflection. — So long as sound-waves are not ob-
structed in their motion they are propagated in the form of
concentric spheres ; but when they meet with an obstacle,
they follow the general law of elastic bodies; that is, they
return upon themselves, forming new concentric waves, called
reflected waves, which seem to emanate from a second center
on the other side of the reflecting body. This phenomenon
is called the reflection of soimd-iuaves. A (Fig. 160) repre-
sents a vibrating particle or a sonorous center from which

Fig. 160.

emanates a series of waves. P Q represents an obstacle with
a flat surface turned toward the waves. Take, for example,
the incident wave MCDN, emitted from the center A; the
corresponding reflected wave is represented by the arc C K D

198 MOLAR DYNAMICS.

of a circle whose center a is as far beyond the obstacle P Q
as A is in front of it.

Join any point, C, of the reflecting surface to the sonorous
center, and the line A C represents one of an infinite number
of directions in which energy is transmitted by a sound-wave.
Such a line may conveniently be called a sound-ray. Let fall
the line H C normal to the surface at the point of incidence C.
The angle A C H is called the angle of incidence. The ray
A C after reflection takes the direction C B, which is a prolon-
gation of «C. The angle BCH is called the angle of reflection.
An observer at B receives sound-waves not only directly from
A in the line AB, but also from C in the line CB. Hence he
hears two sounds, one (to speak in common parlance) proceed-
ing from point A, and the other from point C. The latter
travels from A to C and from C to B, a longer distance than
AB, and is therefore heard later than the former. If the
interval of time between their arrivals at B be greater than
about a fifth of a second, the ear is able to separate the two
sensations and the latter appears as an eclio. If the interval
of time be too short, then only a single and perhaps somewhat
blurred and indistinct sound is heard. The latter phenomenon
is usually called resonance. Such an effect is experienced
frequently by a person listening to his own voice in a large
hall.

If the obstacle PQ present a concave surface, the wave-
front after reflection will be less convex, and may become
plane or even concave according to the degree of the concavity
of the reflector and the position of the sounding body.

165. Sound-waves reflected by concave mirrors.

Experiment. — Place a watch at the focus A (Fig. 161) of a concave
mirror G. At the focus B of another concave mirror H, place the large
opening of a small tunnel, and with a rubber connector attach the bent
glass tube C to the nose of the tunnel. The extremity D being placed
in the ear, the ticking of the watch can be heard very distinctly, as though

REFRACTION OF SOUND-WAVES. 199

it were somewhere near the mirror H. Though the mirrors be 12 feet
apart, the sound will be louder at B than at an intermediate point E.

How is this explained ? Every air-particle in a certain

radial line, as A.c, receives and transmits motion in the

direction of this line ; the last particle strikes the mirror

at c, and bein^ perfectly , ^7

G ^jC- ^ H

elastic, bonnds off in the AT

2|a

V .If y

Fig. 161.

direction c c', commnnicat-
ing its motion to the par-
ticles in this line. At c'
a similar reflection gives
motion to the air particles
in the line c'B. In consequence of these two reflections, all
divergent sound-rays, as Ad, Ae, etc., that meet the mirror G,
are there rendered parallel, and afterwards rendered conver-
gent at the mirror H. The practical result of the concentra-
tion of this scattering energy is, that a sound of great intensity
is heard at B. The points A and B are called the foci of the
mirrors. The front of the wave as it leaves A is convex, in
passing from G to H it is plane, and from H to B concave.
If you fill a large circular tin basin with water, and strike
one edge with a knuckle, circular waves with concave fronts
will close in on the center, heaping up the water at that point.

Long "whispering-galleries" have been constructed on this
principle. Persons stationed at the foci of the concave ends
of the long gallery can carry on a conversation in a whisper
which persons between cannot hear. The external ear is a
wave-condenser. The hand held concave behind the ear, by
its increa'sed surface, adds to its efficiency.

166. Refraction. — If you place your ear at the small end
of a tunnel, C (Fig. 162), and listen to the ticking of a watch,
A, about 4 meters distant, and then introduce a collodion
balloon, B, filled with carbonic acid gas between your ear

200

MOLAE DYNAMICS.

and the watch, and very near the latter, the sound becomes
louder.

The cause is obvious : for let the curved lines a, h, c, etc.,
represent sections of sound-waves with convex fronts, and B

Fig. 162.

a spherical body of carbonic acid gas which is denser than air ;
then it is clear that, owing to the slower progress of the waves
in the denser gas, they would become flattened on entering
this gas, and the waves of convex fronts may be changed to
waves of plane fronts. Again, points at the extremities of
the waves, having less distance to travel in the denser gas
than points near the center, would emerge first and get in
advance, and thus the wave fronts which are plane or nearly
so while wholly in the dense gas, become concave on leaving
it. By these changes in the form of the wave fronts, sound
energy which was originally becoming diffused through wider
and wider space, and therefore becoming less intense as it
progressed, is so changed in direction in passing into and out
of a medium of greater density, that the energy is finally
concentrated at a distant point, as at C, and thereby inten-
sified.

Any change in direction of sound, caused by passing from
a medium of a certain density into a medium of different
density, is called refraction.

167. Diffraction. — When sound-waves encounter an ob-
stacle, a series of secondary waves are formed with the edges

REENFORCEMENT OF SOUND-WAVES.

201

of tlie obstacle as centers. These waves appear to flow around
behind the object, so that the obstacle is able to produce only
a partial sound-shadow. ^'Sounds heard around a corner"
are thus accounted for. This bending, as it were, of waves
around an obstacle is called diffraction.

Section V.

REENFORCEMENT OF SOUND-WAVES ; INTERFERENCE OF
SOUND-WAVES.

168. Reenforceynent of sound-waves.

Experiment 1. — Set a diapason in vibration ; you can scarcely hear
the sound unless it is held near the ear. Press the stem against a table ;
the sound rings out loud, but the waves seem to proceed from the table.

When only the fork vibrates, the prongs, presenting little
surface, cut their way through the air, producing very slight
condensations, and consequently waves of little intensity.
When the fork rests upon the table, the vibrations are com-
municated to the table ; the table with its larger surface
throws a larger mass of air
into vibration, and thus
greatly intensifies the sound-
waves. The strings of the
piano, guitar, and violin owe
as much of their loudness of
sound to their elastic sound-
ing-boards as the fork does
to the table.

169. Reenforcenient hy bod-
ies of air ; resonators.

— e,

a — :.':::

a""

Fig. 163.

Experiment 2. — Take a glass
tube, A (Fig. 163), 16 inches long
and 2 inches in diameter ; thrust one end into a vessel of water, C, and
hold over the other end a vibrating diapason, B, that makes (say) 256

202 MOLAR DYNAMICS.

vibrations in a second. Gradually lower the tube into the water, and
when it reaches a certain depth, i. e. when the column of air o c attains a
certain length, the sound becomes very loud ; as the tube is lowered
below this point, the sound rapidly dies away.

Columns of air, as well as sounding-boards, serve to reenforce
sound-waves. The instruments which enclose the columns of
air are called resonators. Unlike sounding-boards, they can
respond loudly to only one tone, or to a few tones of widely
different pitch.

How is this reenforcement effected? When the prong a
moves from one extremity of its arc a' to the other a", it
sends a condensation down the tube ; this condensation,
striking the surface of the water, is reflected by it up the
tube. Now suppose that the front of this reflected conden-
sation should just reach the prong at the instant it is starting
on its retreat from a^^ to a' ; then the reflected condensation
will conspire with the condensation formed by the prong in
its retreat to make a greater condensation in the air outside
the tube. Again, the retreat of the prong from a^^ to <x'
produces in its rear a rarefaction, which also runs down the
tube, is reflected, and will reach the prong at the instant it is
about to return from ct' to a", and to cause a rarefaction in its
rear ; these two rarefactions moving in the same direction
conspire to produce an intensified rarefaction. The original
sound-waves thus combine with the reflected, to produce
resonance 5 but this can happen only when the like parts of
each wave coincide each with each ; for if the tube were
somewhat longer or shorter than it is, it is plain that conden-
sations and rarefactions would meet in the tube, and tend to
destroy each other.

The loudness of sound of all wind instruments is due to
the resonance of the air contained within them. A simple
vibratory movement at the mouth or orifice of the instrument,
scarcely audible in itself (such as the vibration of a reed in

MEASURING LENGTHS OF SOUND-WAVES.

203

reed pipes, or a pulsatory movement of the air, produced by
the passage of a thin sheet of air over a sharp wooden or
metallic edge, as in organ pipes, flutes, and flageolets, or more
simply still by the friction of a gentle stream of breath from
the lips sent obliquely across the open end of a closed tube
or pen-case), is sufficient to set the large body of enclosed air
in the instrument into vibration, and the sound thus reenforced
becomes audible at long distances.

Experiment 3. — Attach a rose gas-burner, A (Fig. 164), to a metal
gas-tube about 1 m in length, and connect this by a
rubber tube with a gas-nipple. Light the gas at the
rose burner, and you will hear a low, rustling noise.
Remove the conical cap from the long tin tube (Fig.
158), support the tube in a vertical position, and gradu-
ally raise the burner into the tube ; when it reaches a
certain point not far up, the body of air in the tube will
catch up the vibrations, and give out deafening sound-
waves that will shake the walls and furniture in the
room.

170. Measuring ivave-lengths and the speed
of sound-iuaves. — Experiments like that de-
scribed on p. 201 enable us readily to meas-
ure the length of the wave produced by a fork
whose vibration number is known, and also to
measure the velocity of sound-waves. It is
evident that if a condensation generated by
the prong of the fork in its forward movement from «' to «"
(Fig. 163) meet with no obstacle, its front, meantime, will
traverse the distance od, or twice the distance oc ; hence the
length of the condensation is the distance od. But a conden-
sation is only one-half of a wave, and the passage of the
prong from a' to a" is only one-half of a vibration; conse-
quently the distance oc? is one-half of a wave-length, and the
distance o c is one-fourth of a wave-length. The measured dis-
tance of oc in this case is about 13.13 inches ; hence the

Fig. 164.

204

MOLAR DYNAMICS.

length of wave produced by a C'-fork making 256 vibrations

in a second is (13.13 inches X 4 =) 52.5 inches = 4.38 feet.

And since a wave from this fork travels 4.38 feet in -^^^ of a

second, it will travel in an entire second (4.38 feet X 256 =)

1121 feet. The distance oc varies with the temperature of

the air.

It is evident that the three quantities expressed in the

formula

velocity

wave-length = — rr — -.

number oi vibrations

bear such a relation to one another that if any two be known,
the remaining quantity can be computed. It will further be
observed that ivith a given velocity the ivave-length varies
inversely as the number of vibrations ; i.e. the greater the num-
ber of vibrations per second, the shorter the wave-length.
171. Interference of soimd-ivaves.

Experiment 4-

Hold a vibrating diapason over a resonance-jar, as in
Fig. 165. Koll the diapason over
slowly in the fingers. At certain
points a quarter of a revolution
apart, when the diapason is in an
oblique position with reference to
the edge of the jar as represented
in the figure, the reenforcement
from the tube almost entirely dis-
appears, but it reappears at the in-
termediate points. That is, there
are four intervals in the space
around the fork where the two
series of waves generated by the
two tines interfere to produce
mutual destruction. These are'
called technically the cones of silence. Return to the position where
there is no resonance, and enclose in a loose roll of paper the prong
farthest from the tube, without touching the diapason, so as to prevent
the sound-waves produced by that prong from passing into the tube; the

FORCED AND SYMPATHETIC VIBRATIONS.

205

resonance resulting from the vibrations of the other prong immediately
appears.

^Experiment 5. — Select two of the tubes (Fig. 188) of nearly the same
length, blow through them, and notice the peculiar throbbing sound
produced by the interference of the two sounds.

Experiment 6. — Stop one of the orifices of a bicyclist's
whistle (Fig. 166), and sound one whistle at a time. The
sound of each is clear and smooth. Sound both whistles at
the same time, and you obtain the usual rough and discord-
ant sound.

The two whistles of unequal length give out waves of
slightly different length, so that at certain short intervals
both waves will interfere in the same phase (i. e. condensation
with condensation) and produce intensified sounds which
are heard at long distances, while at other intervals they
interfere in opposite phases (i.e. condensation with rarefac-
tion), and the result of their mutual destruction is to cause
the otherwise smooth sound to become broken or rattling.

Two sound-waves may combine to produce a sound louder

Fig. 166.

or weaker than either alone would produce^ or may even cause

silence. This combination of sound-waves to produce a louder or weaker

sound is called interference.

172. Forced and sympathetic vibrations.

Experiment 7. — Suspend from a frame several pendulums. A, B, C,
etc. (Fig. 167). A and D are each 3 feet long,
C is a little longer, and B and E are shorter.
Set A in vibration, and slight impulses will be
communicated through the frame to D, and
cause it to vibrate. The vibration-period of D
being the same as that of A, all the impulses
tend to accumulate motion in D, so that it soon
vibrates through arcs as large as those of A.
On the other hand, C, B, and E, having differ-
ent rates of vibration from that of A, will at
first acquire a slight motion, but soon their
vibrations will be in opposition to those of A,

and then the impulses received from A will tend to destroy the slight

motion they had previously acquired.

Experiment 8. — Press down gently one of the keys of a piano so as to

206

MOLAR DYNAMICS.

Fig. 168.

raise the damper without making any sound, and then sing loudly into
the instrument the corresponding note. The string corresponding to this
note will be thrown into vibrations that can be heard for several seconds
after the voice ceases. If another note be sung, this string will respond
only feebly.

Eaise the dampers from all the strings of the piano by pressing the
foot on the right-hand pedal, and sing strongly some note into the piano.
Although all the strings are free to vibrate, only those will respond loudly
that correspond to the note you sing, i.e. those that are capable of making
the same number of vibrations per second as are produced by your voice.
Experiment 9. — Take two forks, A and B (Fig. 168), tuned exactly in

unison, and mounted on reso-
nance-boxes, and place them
from three to ten meters apart.
Fasten, by a bit of sealing-wax,
a thread to a thin piece of glass
12 mm square (glass used for
microscopic mountings is the
best, or a piece of photographic
tintype plate will answer well),
and suspend so as to touch a
corner of one of the prongs of the fork B. Set the fork A in vibration
by drawing a resined bass-viol bow strongly across the ends of its prongs.
In about ten seconds stop the vibrations of A with the fingers, and you
will see and hear the piece of glass rattling against the prong of the fork
B ; remove the glass, and place the ear near the fork B, or better, the
open end of the box, and you may hear a distinct sound, showing that
the fork B has been thrown into a state of vibration by the fork A.

So the pulses that traverse the air between the forks, so
gentle that only the sensitive organ of the ear can perceive
them, become great enough to move the rigid steel when the
energy of their blows, dealt at the rate of perhaps 512 in a
second, accumulates. The large number of blows makes up
for the feebleness of the individual blows.

These experiments show that a vibrating body tends to
make other bodies near it vibrate, even if their periods of
vibrations be different. Vibrations of this kind, such, for
example, as those of B, C, and E in Exp. 7 and those

PITCH OF MUSICAL SOUNDS.

207

generated in the sounding-boards of pianos, violins, etc., are
called forced vibrations. But if the period of the incident
waves of air be the same as that of the body which they
cause to vibrate, the amplitude and intensity of the vibrations
become very great, like that of the pendulum D, and those of
the piano strings which gave forth the loud sounds. Such
are called syvipathetic vibrations.

Section VI.

PITCH OF MUSICAL SOUNDS.

173. On ivhat pitch depends.

Exjperiment 1. — Draw tlie finger-nail or a card slowly, and then
rapidly, across tlie teetli of a comb. The two sounds produced are com-
monly described as low or grave, and high or acute. The hight of a
musical sound is its pitch.

Experiment 2. — Cause the circular
sheet-iron disk A (Fig. 169) to rotate,
and hold a corner of a visiting-card so
that at each hole an audible tap shall be
made. Notice that when the separate
taps or noises cease to be distinguishable,
the sound becomes musical ; also, that
the pitch of the musical sound depends
upon the rapidity of the rotation, i.e.
upon the frequency of the taps.

Experiment 3. — Hold the orifice of a
tube B so as to blow through the holes
as they pass. When rotating slowly,
separate puffs, from which it hardly
seems possible to construct a musical
sound, are heard. When, however, the
ear is no longer able to detect the sepa-
rate puffs, the sound becomes quite musi-
cal, and the pitch rises and falls with the
speed.

Fig. 169.

Pitch depends upon the number of sound-waves striking the

208 MOLAR DYNAMICS.

ear per second. If the source of the sound-waves and the receiv-
ing ear he both stationary, the pitch depends upon the frequency
of vibration, or wave-length; i.e. the greater the number of
vibrations per second, or the shorter the wave-length, the higher
the pitch.

Since pitch depends upon the number of sound-waves strik-
ing the ear per second, a sound must rise in pitch if we
rapidly approach the source of the sound-waves, or the source
rapidly approach us, as evidently more sound-waves will then
strike the ear per second than otherwise would happen. The
pitch of the whistle rises on the rapid approach of a locomo-
tive, and falls again as the engine travels away.^

1 It may be of interest to consider more in detail two cases of relative motion
between the ear and the sounding body :

Case I. Source stationary and ear approaching it with velocity v^ (Fig. 170) per

Fig. 170.

second. Let v = the velocity of the sound-waves per second, and n = the number of
vibrations made by the vibrating body {e.g. a fork) per second. The wave-length is

- and the number of waves in the space v-y is t\ -f- - or — *- • The number of waves

tone

striking the ear per second is, therefore, n + — ■— , or w I 1 + ^1. If Vi= v, the

perceived is an octave above the normal pitch of the fork. If v^ = r, but the motion
of the ear be away from the fork, this quantity becomes n (1—1), or zero, and no
sound-waves reach the ear.

Case II. Ear stationary, but the fork moves towards it with a velocity of v^ per
second. While the fork moves from A to B (Fig. 171), the first sound-wave sent out

from the fork at the beginning of the second has travelled all the way from A to C.
The last wave sent out in that second of time is just starting from B when the very
first wave has reached C. Thus there are n waves between B and C, and the length

VIBRATION-FREQUENCY OF A TONE. 209

174. Hoiv to find the vibration-frequency of a tone. — The
siren. — The perforated wheel described above is a cheap
imitation of a portion of an important instrument called a
siren. The instrument complete has an attachment called a
counter^ which shows the number of revolutions the wheel
makes in a given time.

Suppose that it is required to ascertain the number of
vibrations per second necessary to produce a given pitch.
Take some instrument that gives the required pitch, e.g. a
tuning-fork, and set it in vibration ; also rotate the siren,
causing the pitch of its sound gradually to rise until it corre-
sponds with the pitch of the fork ; then, sustaining that
pitch, set the counter in operation, and at the end of a given
time read off the number of revolutions made by the wheel ;
this number multiplied by the number of holes in the wheel
gives the number of sound-waves produced by the wheel
during the given time, and the number of vibrations made by
the fork in the same time ; and this number divided by the
number of seconds employed gives the number of vibrations
that must be made in a second by any instrument in order to
produce a sound of the same pitch. With the siren we may
even determine the number of vibrations made by the wing of
a fly which buzzes around our ears.

The vibration frequency of a fork may be easily found by means
of an apparatus called a vibrograph. One of the tines of the fork a
(Fig. 172) has a small elastic indicator attached to its extremity.
The sharp point of this indicator touches a smoked glass plate, k,
below. Above the glass plate is suspended a pendulum with a
heavy bob. Beneath the bob is another indicator which just grazes

of each is only ^' and the number of vibrations received by the ear per second is

V -^ ^' or — This shows that when Vn= v, this fraction has the value oo ;

n v — Vz

also that in order to get the octave above the normal, v^ must equal -•

It is obvious that in Case 11. there is a shortening of the waves, but there is no
shortening in Case I.

210

MOLAR DYNAMICS.

the glass as it passes the lower part of its arc. The experimenter
first finds the exact fraction of a second occupied by the pendulum
in making one complete or double vibration. The fork is then put
in vibration and the block h carrying the glass plate is drawn along

Fig. 172.

beneath the style, which marks upon the glass a wave line. Imme-
diately after the glass is put in motion the pendulum is set swinging
and allowed to traverse the plate width- wise three times, making,
with its indicator, three lines athwart the wave line. Now the
interv^ of time between the instants when the first and the third
of these lines are made is the time of one complete vibration. The
number of vibrations which the fork made in this interval may be
determined from the sinuous curved line intervening between the
lines made by the pendulum. The number of vibrations made by the
fork in a certain fraction of a second having been ascertained in this
manner, the vibration number per second is calculated therefrom.

175. Distmction between 7ioise and 'musical sound. — If the
body that strikes the air deal it but a single blow, like the
discharge of a fire-cracker, the ear receives but a single shock,
and the result is called a noise. If several shocks be slowly
received by the ear in succession, the ear distinguishes them
as so many separate noises. If, however, the body that strikes
the air be in vibration, and deal it a great number of little
blows in a second, or if a large number of fire-crackers be

MUSICAL SCALE. 211

discharged one after another very rapidly, so that the ear is
unable to distinguish the individual shocks, the effect produced
is that of one continuous sound, which may be pleasing to the
ear ; and, if so, it is called a musical sound. But continuity
of sound does not necessarily render it musical. The sound
produced by a hundred children beating various articles in a
room with clubs might not be lacking in continuity, but it
would be an intolerable noise. There would be wanting those
elements that please the ear ; viz. regularity both in perio-
dicity and intensity of the shocks which it receives. The
distinction between music and noise is, generally speaking,
a distinction between the agreeable and the disagreeable,
between regularity and confusion.

176. Musical scale. — Suppose a body, e.g. a tuning fork,
to make 261 vibrations per second, the sound produced is

recognized by our musical sense as the note ^ T^ which

corresponds with the so-called middle C (c', or French utg) of
a piano tuned to the national standard pitch. ^

The pitch of a sound produced by twice as many vibrations
as that of another sound is called the octave of the latter.
Between two such sounds the voice rises or falls, in a manner
very pleasing to the ear, by a definite number of steps called
micsical intervals. This gives rise to the so-called diatonic
scale, or gamut. Long before any one had attempted to find
the frequency of vibration of a sounding body, men had used
a succession of sounds, differing in pitch, determined only by
their musical sense and not by arbitrary agreement. The
number of vibrations which shall constitute a given note is
purely arbitrary, and differs slightly in different countries ;

1 In a conTention of piano manufacturers held in New York it was decided that
the national pitch to go into effect July 1, 1892, should he the standard French,
Austrian, and Italian pitch of 435 (Ag) double vibrations in a second at 68° F.

212 MOLAR DYNAMICS.

but the ratios between the vibration numbers of the several
notes of the gamut and the vibration number of the first or
fundamental note of the gamut, are the same among all
enlightened nations.

The successive tones of the diatonic scale of C are related
to one another Vv^ith respect to vibration frequency as follows :

-&- — ^-

of'

Uts

res

mis

fas

sols

las

sis

Ut4

261

293.62

326.25

348

391.5

435

489.37

522

256

: 288

: 320 :

341.3

: 384 :

426

: 480 :

512

: f

: 1 :

: 1 '

-i

: -V- :

No. of vi-
brations

Ratios

The ear is wholly incapable of determining the number of
vibrations corresponding to a given tone, but it is capable of
determining with wondrous precision the ratio of the vibration
numbers of two notes ; hence all music must depend upon the
recognition of such ratios, and for this reason the vibration
ratios given above are of the utmost importance. An octave
below c' is c ; two octaves below, Ci, and so on. In a similar
manner the octaves below any other tone are indicated.

The following are some of the various musical intervals occurring
w^ithin the diatonic scale : Minor second^ e' : f^ or V : of' : : 15 : 16 ;
Major second, of : d', f : g', or a' : b' : : 8 : 9 ; Minor third, e' : g' or
a' : c^' : : 5 : 6 ; Major third, of : e', f ' : a', or g' : b' : : 4 : 5 ; fourth,
of \l', d' : g', e' : a', or g' : c'' : : 3 : 4 ; jifth, of : g', e' : b', or f : Q," : :
2 : 3, etc.

177. Limits of scale and of audibility. — The lowest note
of a 7-J octave piano makes about 27-|- vibrations per second;
the highest, about 4,224 vibrations per second ; but these
extreme notes have little musical value, and the lowest notes
are chiefly used for their harmonics only (see p. 220).
The range of the human voice lies between 61 and 1305

LIMITS OF SCALE AND OF AUDIBILITY. 213

vibrations per second/ or a little more than three octaves ; an
ordinary singer has about the compass of two octaves.

The ear is capable of hearing vibrations far exceeding in
number the requirements of music. It can appreciate sounds
arising from 32 to 38,000 vibrations ^ per second, i.e. a range
of about eleven octaves, and a corresponding range of wave-
length between seventy feet and three or four tenths of an
inch. These numbers vary considerably, however, with the
person. Exceptional ears can hear as many as 50,000 vibra-
tions. Some ears can hear a bat's cry, or the creaking of a
cricket; others cannot. Singing mice are sometimes placed
on exhibition. Of those who go to hear them, some can hear
nothing, others a little, and others again can hear much. In
the ability to hear sharp sounds, no animal is superior to the
cat, which finds her prey in the dark by its squealing. High
tones are heard with difficulty in the presence of low ones.
A lower tone tends to drown a higher one.

Exercises.

1. Find the vibration number for each note of the scale of which c'' is
the first note.

2. What is the vibration number of c an octave below c' ?

3. Find the wave-length corresponding to each note of the scale of
which c' is the first, when the temperature of the air is 16° C. ?

4. Find the length of a resonance tube (disregarding its diameter)
closed at one end, which will respond to c" when the temperature is
16° C. ?

5. a. The interval between e' and c'' is called a minor sixth ; what is
the vibration ratio for this interval ? 6. What is the note a minor sixth
above a' ?

6. Make out a series of fractions which shall express the vibration
ratios of each tone in the diatonic scale c'', d'^, etc., as compared with
c'; i.e. continue the series of ratios given on p. 212 through another
octave.

1 Pietro Blaserna, in his " Theory of Sound."

2 Preyer places the lowest limit for some ears at 16 vibrations per second.

214 MOLAK DYNAMICS.

7. What is the vibration number of a' in a scale in which c' (the key-
note) = 256 vibrations ?

8. The same singer may not be able to sing twice alike, i.e. in the
same key ; how is it possible that the singing in both instances may be
equally correct ?

9. If one ear can hear a certain sound at 5 feet from the sounding
body, and another ear at only 3 feet, how many times more sensitive is
the former ear than the latter ?

10. Why does the same bell always give a sound of nearly the same
pitch ?

11. a. What is the effect of striking a bell with different degrees of
force ? b. What change in the vibrations is produced ? c. What property
of sound remains the same ?

12. a. Strike a key of a piano and hold it down ; what is the only
change you observe in the sound produced, while it remains audible ?
6. What is the cause of this change ?

Section VII.

COMPOSITION OF SONOROUS VIBRATIONS AND THEIR RESULTANT
WAVE-FORMS.

178. Coexistence and siqjerjjosition of ivaves. — Interference.
— When two or more currents of waves traverse the same
medium at the same time and in the same direction, so that
one set of waves is, as it were, superposed upon another, there
are imparted to every particle of the medium simultaneously
all the vibratory motions peculiar to the several waves. When
two or more systems of waves act on a particle at the same
time, they are said to interfere. The resultant motion of any
particle at a given instant would be found on the principle of
parallelogram of motions ; or, in case the several motions are
parallel and occur in the same time, the resultant is the
algebraic sum of the several motions. This will be best
understood by means of graphical representations. In A
(Fig. 173) are represented by dotted lines the wave lines of
two coexisting currents of waves having the same wave-length

COEXISTENCE AND SUPERPOSITION OF WAVES. 215

and phase, but the amplitude of one greater than that of the
other. For example, the amplitudes of the vibrations for the
particle a are respectively a c
and a e. Their algebraic sum
is ad. In like manner the
displacement of any particle
of the medium traversed by
the several wave currents at
any instant is determined.
The heavy line represents
the form of the joint wave
resulting from the combina-
tion of the two. It will be
seen that the only change is
one of amplitude or inten-
sity.

. In B are two wave-cur-
rents whose waves are of
the same length and ampli-
tude, but with a difference

of phase of 1 of a period, '^~-'' '"-^ ^^-''' ^^— ''

. ^ . * ^ P Fig. 173.

I.e. one is a quarter oi a

wave-length behind the other. The result is a wave of the

same length, but of different phase and amplitude.

In C are two sets of waves of like length, of different
amplitudes, and of opposite phases, or one is half a wave-
length behind the other. The result is a set of waves of the
same length, but diminished intensity. In D the conditions
are the same as in C except that the components have the
same amplitude. The result is that the two components de-
stroy each other, and the particles of the medium are undis-
turbed, as indicated by the straight line.

In A (Fig. 174) are given two wave-currents whose wave-
lengths are as 1 : -J- and whose phases in the beginning agree.

216

MOLAR DYNAMICS.

The resultant of this combination with still another of ^ the
wave-length of the longest is shown in B. In C is the same

Fig. 174.

combination as in A, but the phases differ by ^ of a period of
the shorter wave.

Fig. 175 represents wave-lines drawn by a vihrograph. The
second line represents a sound two octaves above that which

2 \AAAA/VV\AAAAAAAAAAAAAAAAAAAA^AAAAAAAA/\AA/\AAAAAA/\AA/\/\AAAAA/^^

Fig. 175.

the first line represents, and the third line shows the result
that is produced by causing the fork to have two sets of
simultaneous vibrations.

Extend the arifi horizontally and cause the hand to move from
side to side throuofh a wide arc and at the same time cause the

COEXISTENCE AND SUPERPOSITION OF WAVES. 217

hand to move in the same plane through shorter arcs in a sort of
jerky movement. The result of these combined motions bears some
faint resemblance to the motion of the tine as it draws line 3 above,
or to the motion of an air particle as the tv^o currents of waves
generated by the fork pass it.

One may see a typical representation of superposed waves, or
currents of small waves and wavelets creeping over the backs of
larger ones and carving their surfaces into ragged and ever-changing
outlines, in watching the billows of the sea, especially when the
surface is swept by a breeze of varying intensity.

In the diagrams given above only transverse vibrations are
represented, bnt the results there depicted apply equally well
to longitudinal vibrations and waves of condensations and
rarefactions. In Fig. 176 the heavy line AB is a tyiykal
representation of the resultant of two currents of aerial

Fig. 176.

sound-waves an octave apart, while the rectangular diagram
C D is intended to represent a portion of a transverse section
.of a body of air traversed by the joint wave corresponding to
the heavy wave-line above. The depth of shading in different
parts indicates the degree of condensation or rarefaction at
those parts.

218

MOLAR DYNAMICS.

Section YIII.

VIBRATION OF STRINGS.

179. Sonometer. — This instrument consists of two or more
piano-wires of different thicknesses stretched lengthwise over
a resonance box. One end of each wire is attached to the
shorter arm of a bent lever, A or B (Fig. 177), and the tension

Fig. 177.

of the wire is regulated both by the lengths of the longer
arms employed and by the magnitude of the weights suspended
therefrom. The length of the vibrating portion of the strings
is regulated by the sliding bridge C.

Experiment 1. — Remove the bridge C, pluck one of the strings with
the fingers at the middle point, causing it to vibrate as a whole, and note
the pitch of the sound. Place the bridge under the same wire, and move
it gradually toward one end of the sonometer, thereby shortening the
vibrating portion ; the pitch rises as the vibrating portion is shortened.
Vary the position of C until a pitch is obtained an octave above the pitch
given at first when the entire wire was vibrating. It will be found that
the length of the wire which gives the higher note is just half the original
length; i.e. hy halving the wire its vibration-numher is doubled. At two-
thirds its original length, it gives a note at an interval of a fifth above
that given by its original length ; and generally the reciprocals of the
fractions (p. 212) representing the relative vibration-numbers of the
several notes of a scale represent the relative lengths of the wires that
produce these notes.

Now increase the tension of the wire ; the pitch rises. Increase the
tension until the pitch has risen an octave; it will be found that the
tension has been increased fourfold.

STATIONARY VIBRATIONS, NODES, ETC. 219

Next try two wires whose lengths and tension are the same, but whose
diameters are (say) as 1 : 2, and whose masses are consequently as 1 : 4 ;
the pitch given by the wire of greater mass is an octave lower than the
pitch given by the other wire.

These conclusions may be summarized thus : The vibration-
numbers of strings of the same material vary inversely as their
lengths and the square roots of their masses per unit length, and
directly as the square roots of their tensions.

180. Stationary vibrations, nodes, etc.

Experiment 2. — Hold one end of a rubber tube about 2 m long, while
the other is fixed, and send along it a regular succession of equal pulses
from the vibrating hand ; it will be easy, by varying the tension a little,
to obtain a succes- ^ c

sion of gauzy spindles ^ ^^,--'^*"™'llli^..^^^^^

(Fig. 178) separated 111^"^'^^.

by points that are

1 V ^ ^- Fig. 178.

nearly or quite at rest.

Unlike the earlier experiments, the waves here do not appear to travel

along the tube ; yet in reality they do traverse it. The deception is

caused by stationary points being produced by the interference of the

advancing and retreating waves.

This interference of direct and reflected waves gives rise
to an important class of phenomena called stationary vibra-
tions. The points of least motion, as a, b, and e (Fig. 178), are
called nodes (from fancied resemblance to knots) ; the points
of greatest amplitude, as d and c, are called antinodes ; and the
portions between the nodes are called venters.

In a similar manner a string may be made to vibrate in 3,
4, etc., parts, as shown in C, D, and E (Fig. 179). The pitch
of the tone produced by a string when it vibrates as a whole,
as in A, is called the fundamental pitch of the string. The
vibration frequency when the string divides into halves, as
in B, is twice as great and consequently the pitch of the tone
produced is an octave above that of the fundamental. Gen-
erally the vibration frequency varies as the number of venters
into which the string divides.

220

MOLAR DYNAMICS.

Tones produced by a string or other body that vibrates in
parts are called overtones or partial tones. If the overtones

Fig. 179.

harmonize (p. 223) with the fundamental of the vibrating
body, they are called harmonics.
181. Complex vibrations.

Experiment 3. — Strike one of the lowest notes of a piano, hold the
key down, and immediately apply the tip of the finger to some point of
the wire struck, and notice any changes in tone that may occur after
applying the finger. Kepeat this at many points along the string. If the
fundamental sound disappear, there will probably be a sound of a higher
pitch that will continue, showing that although you have stopped one set
of vibrations, there were still other vibrations in the string of a higher
vibration-period which you did not stop, and which now become audible
since the louder fundamental is silenced.

Experiment 4. — Press down the C'-key gently, so that it will not
sound ; and while holding it down, strike the C-key strongly. In a few
seconds release the key, so that its damper will stop the vibrations of the
string that was struck, and you will hear a sound which you will recog-
nize by its pitch as coming from the C'-wire. Place your finger lightly on
the C'-wire, and you will find that it is indeed vibrating. Press down the
right pedal with the foot, so as to lift the dampers from all the wires,
strike the C-key, and touch with the finger the C'-wire; it vibrates.
Touch the wires next to C, viz. B and T>' ; they have only a slight forced
vibration. Touch G' ; it vibrates.

ISTow it is evident that the vibrations of the C and G'-wires
are sympathetic. But a C-wire vibrating as a whole cannot
cause sympathetic vibrations in a C'-wire ; but, if it vibrates

TONES AND NOTES. 221

in halves, it may. Hence, we conclude that when the C-wire
was struck it vibrated, not only as a whole, giving a sound of
its own pitch, but also in halves ; and the result of this latter
set of vibrations was, that an additional sound was produced
by this wire, just an octave higher than the first-mentioned
sound.

Again, the G'-wire makes 391.5 vibrations in a second, or
three times as many (130.5) as are made by the C-wire ; hence
the latter wire, in addition to its vibrations as a whole and in
halves, must have vibrated in thirds, inasmuch as it caused
the G'-wire to vibrate. It thus appears that a string may
vibrate at the same time as a whole, in halves, thirds, etc.,
and the result is that a sensation is produced that is com-
pounded of the sensations of several sounds of different pitch.
A sound so simple that it cannot be resolved (see p. 227) is
called a tone.

182. Tones and notes. — A sound composed of many tones
is called a note.

Not onl}?- do stringed instruments produce notes, but no
ordinary musical instrument is capable of producing a simple
tone, i.e. a sound generated by vibrations of a single period.
In other words, ivhen any note of any musical instrument is
sounded, the7'e is produced, in addition to the primary tone, a
number of other tones in a progressive series, each tone of the
series being usually of less intensity than the preceding.^ The
primary or lowest tone of a note is usually sufficiently intense
to be the most prominent, and hence is called the fundamental
tone.

Strings when struck produce many overtones, according to
the place where they are struck, the nature of the stroke, and
the density, rigidity, and elasticity of the string.

1 The so-called Fourier theorem translated from the language of kinematics into
that of acoustics asserts that " every regular musical sound is resolvable into a
definite number of simple tones whose relative pitch follows the law of the partial-
tone series."

222

MOLAR DYNAMICS.

183. Beats.

Experiment 5. — Strike simultaneously the lowest note of a piano and
its sharp (black key next above), and listen to the resulting sound.

You hear a peculiar wavy or throbbing sound, caused by an
alternate rising and sinking in loudness. This phenomenon
is still more conspicuous where the two lowest adjacent notes
of a large organ are sounded together. Each recurrence of
the maximum intensity is called Si beat.

Let the continuous curved line A C (Fig. 180) represent a
series of waves caused by striking the lower key, and the

Fig. 180.

dotted line a series of waves proceeding from the upper key.
Now the waves from both keys may start together at A ; but
as the waves from the lower key are given less frequently, so
are they correspondingly longer ; and at certain intervals, as
at B, condensations will correspond with rarefactions, pro-
ducing by their interference momentary silence, too short,
however, to be perceived ; but the sound as perceived by the
ear is correctly represented in its varying loudness by the
curved line A'B'C.

It will be apparent from the study of Fig. 180 that exactly
one beat will occur in each interval of time during which the
acuter of two simple tones performs one more vibration than
the graver tone.

Hence the number of beats per second due to two simple tones
is equal to the difference of their respective vibration numbers.

ORIGIN OF HARMONY AND DISCORD. 223

The sensation produced on the ear by such a throbbing sound,

when the beats are sufficiently frequent, is unpleasant, much

as the sensation produced by flashes of light that enter the

eye when you walk on the shady side of a picket fence is

unpleasant. The unpleasant sensation called by musicians

discord is due to beats.

184. Origin of harmony and discord. — The harmonics in

any note are produced successively by two, three, etc., times

the number of vibrations made by its fundamental. Hence,

if any two notes an octave apart, — for instance, C and C, —

be sounded simultaneously, there will result for

C, 1, 2, 3, 4, 5, 6, etc. ) ,. ^. x. , -x. ^- a

^, „ , n , > times the number oi vibrations made
C , 2, 4, 6, etc. )

by the fundamental of C ; so that the fundamental of C, and

its overtones (with the exception of the highest overtones,

which are too feeble to affect the general result), are in perfect

unison with the overtones of C. Not only is there perfect

agreement among the overtones of two notes an octave apart

when sounded together, as when male and female voices unite

in singing the same part of a melody, but the richness and

vivacity of the sound are much increased thereby.

Discord jDroduced by two sounds is explained by the fact
that the sounds produce beats, which do not coalesce because
the interval between them is too long.

As the frequency of the beats increases, a point is finally
reached where they cease to be recognized as distinct
sounds and where they blend into a more or less pure
tone. Beats may thus coalesce to produce beat-tones that
are musical.

It must not, however, be inferred that dissonance disap-
pears immediately upon the intermittences becoming too rapid
for individual recognition. If two tones form a narrower
interval than a minor third, the combined sound is harsh and
grating on the ear.

224 MOLAR DYNAMICS.

Two tones must be in unison to produce absolutely perfect
harmony. The nearest approach to it is an interval of an
octave, and next in rank to the latter is a fifth.

That two notes sounded together may harmonize, it is essential
not only that the pitch of their fundamental tones he so widely
different that they canjiot produce audible beats, but that no
audible beats shall be formed by their overtones, or by an over-
tone and a fundamental.

Eor example, let, the vibration-numbers of the fundamentals
of C and its octave C" be respectively 264 and 528 ; the num-
ber of beats that they give is 264 in a second. If, instead of
C", a note the vibration-number of whose fundamental is 527
be sounded with C, the number of beats produced by their
fundamentals would be 263, and no discord would result
therefrom ; but there would be one beat per second between
the first overtone of C and the fundamental of C", and this
would introduce a discord.

Observe that the relation between the vibration-numbers of
the fundamentals of C and C, C and G, C and F, and C of
any diatonic scale and any note in the same scale, can be
expressed in terms of small numbers, e.g. 1:2, 2:3, 3:4,
etc. (see p. 212). Generally, those notes and only those har-
monize whose fundamental tones bear to one another ratios
expressed by small numbers ; and the smaller the mmibers which
exptress the ratios of the rates of vibration, the more -perfect is
the harmony of two sounds.

Not only may two notes whose relative vibration frequency
is expressible by a simple ratio harmonize, but three or four
may concur with the same result. A sound produced by the
coexistence of three or more notes is called in music a chord.
A consonant chord is a concord ; a dissonant chord is a discord.

Fig. 181 is a graphical representation according to Helmholtz of
the amount of dissonance contained in the several intervals of the
diatonic scale. The intervals reckoned from C, are denoted by dis-

ORIGIN OF HARMONY AND DISCORD.

225

tances measured along the horizontal straight line. The dissonance
for each interval is represented by the vertical distance of the curved
line from the corresponding point on the horizontal line. If we
regard the outline as that of a mountain chain, the discords vv^ould

"be represented by peaks, and the concords by passes, v^hile the
steepness indicates the sharpness of deiinition of the interval. The
calculations on which the curve is based are made from the notes
produced by a violin. For piano-forte notes the curve would be
slightly different.

It follows, from what has been said, that only a limited
number of notes can be sounded with any given note assumed
as a prime without generating discord. Hence, the musical
scale is limited to certain determinate degrees, represented
by the eight notes of the so-called musical or diatonic scale.
This scale is not the result of any arbitrary or fanciful
arrangement, but is composed of notes selected because they
harmonize with the prime of the scale, both as regards their
fundamental tones and their overtones.

Exercises.

1. Prepare a table of the series of overtones of C and G respec-
tively, as on p. 223, and ascertain what overtones of the two series are in
unison.

2. Arrange the notes in a single octave of the diatonic scale in the
order of their rank with reference to their harmonizing with the prime
of the scale, on the principle that "the smaller the numbers which ex-

226 MOLAR DYNAMICS.

press the ratios of vibration, the more perfect is the harmony of two
sounds."

3. Verify your conclusions as follows : Strike the C-key of a piano,
together with each of the seven white keys above it, consecutively, and
compare the results of the different pairs with reference to harmony.

Section IX.

QUALITY OF SOUND.

185. Simple sound-waves can differ only in length and
amplitude; consequently tlie sounds which they produce can
differ only in pitch and loudness. Complex sound-waves may
differ, as we have seen, in /orm, and this gives rise to a prop-
erty of sound called quality (by musicians, timbre). Quality
is that property of sound, not due to pitch or intensity, that
enables us to distinguish one sound from another.

Although the variety of sounds one hears appears well-nigh
infinite, yet no two sounds can differ from each other in any
other respect than pitch, loudness, or quality. The length,
amplitude, and form of the wave completely determine the
wave, and these three elements of a wave are mutually inde-
pendent, i.e. any one may be changed without altering the
other two. Loudness depends on amplitude of vibrations,
pitch on vibration-frequency, and quality on complexity of
the motion of the vibrating particles.

Let the same note be sounded with the same intensity,
successively, on a variety of musical instruments, e.g. a violin,
cornet, clarinet, accordion, jews-harp, etc. ; each instrument
will send to your ear the same number of waves, and the
waves from each will strike the ear with the same force, yet
the ear is able to distinguish a decided difference between the
sounds, — a difference that enables us instantly to identify
the instruments from which they come. Sounds from instru-
ments of the same kind, but by different makers, usually

ANALYSIS OF SOUND-WAVES, 227

exhibit decided differences of character. For instance, of two
pianos, the sound of one will be described as richer and fuller,
or more ringing, or more ''wiry," etc., than the other. No
two huinan voices sound exactly alike.

Section X.

ANALYSIS AND SYNTHESIS OF SOUND-WAVES.

186. Analysis of sound-waves. — The unaided ear is unable,
except to a very limited extent, to distinguish the individual
tones that compose a note. Helmholtz arranged a series of
resonators of brass nearly
spherical in shape, each hav-
ing two openings; one, A
(Fig. 182), large, for the re-
ception of the sound-waves,
and the other, B, small and ^Ijl
funnel-shaped, and adapted
for insertion into the ear.
Each resonator of the series
was adapted by its size to
resound powerfully to only a
single tone of a definite pitch. When any musical sound is
produced in front of these resonators, the ear, placed at the
orifice of any one, is able to single out, from the total number
of tones composing the note, that overtone, if present, to
which alone this resonator is capable of responding. By
applying one resonator after another to the ear a sound is
analyzed into its components. It is thus found, for instance,
that the notes of a clarinet are composed only of the odd
harmonics, or of tones whose vibration numbers are in the
ratios of 1 : 3 : 5 : 7; and that the notes of a flute are substan-
tially those of a tone and its octave. It is found that when a

228 MOLAR DYNAMICS.

note is produced on a given instrument, not 6nly is there a
great variety of intensity represented by the overtones, but
all the possible overtones of the series are by no means
present. Which are wanting depends very much, in stringed
instruments, upon the point of the string struck. For example,
if a string be struck at its middle point, no node can be formed
at that point ; consequently, the two important overtones
produced by 2 and 4 times the number of vibrations of the
fundamental will be wanting. Strings of pianos, violins, etc.,
are generally struck near one of their ends, and thus they are
deprived of only some of their higher and feebler overtones.

" Every vowel is a particular quality of sound." The
mouth cavity acts as a resonator and reflector. To each
vowel corresponds a different form of resonating mouth cavity.
Upon the number of upper overtones which are reinforced,
and the relative intensities of the reinforcement, depends the
quality of the vowel sound produced. Vowel sounds may be
analyzed in a very interesting manner, as follows : Eaise the
cover of a grand piano, press down the "loud pedal," and
sing strongly some vowel sound, projecting the voice upon
the exposed strings. When you cease to sing, that vowel will
be repeated by the strings. Each component of the complex
vibration will be taken up by that string in unison with it,
and by noticing which strings vibrate a qualitative analysis of
the sound is effected,

187. Manomstric flames. — The pitch, intensity, and qual-
ity of a sound may be studied at the same time by causing
sound-waves to impinge directly upon some sensitive body
without any intermediate process of selection. Apparatus
like that shown in Eig. 183 will serve to illustrate.

The cylindrical box A is divided by the membrane a into
two compartments, c and h. Illuminating-gas is introduced
into the compartment <?, through the rubber tube n, and burned
at the orifice d. CD is a frame holding two mirrors, M,

MANOMETRIC FLAMES.

229

placed back to back, so that wMcliever side is turned toward
the flame there is a reflection of the flame.

When the mirror is at rest, an image of the flame will
appear in the mirror as represented by A (Fig. 184). If the
mirror be rotated, the flame appears drawn out in a band of
light, as shown in B of the same figure.

Fig. 183.

Sing into the cone B (Fig. 183) the sound of oo in tool, and
waves of air will run down the tube and beat against the
membrane a, causing it to vibrate. The membrane has im-
pressed upon it a complex motion resembling the original
compound vibration of the vocal chords or other sounding
body. The membrane in turn acts upon the gas in the com-
partment G, throwing it into vibration.

230

MOLAR DYNAMICS.

The result is, that instead of a flame appearing in the rotat-
ing mirror as a continuous band of light, as B (Fig. 184), it is
divided up into a series of tongues of light, as shown in C,

iiMAkkiiMiiiikii

Fig. 184.

each condensation being represented by a tongue, and each
rarefaction by a dark interval between the tongues. The
number and size of the greater tongues indicate the fre-
quency and amplitude of the fundamental vibration ; the

THE PHONAUTOGRAPH OR PHONOGRAPH.

231

subsidiary serrations correspond to the subsidiary vibrations
or overtones.

If a note an octave higher than the last be sung, we obtain,
as we should expect, twice as many tongues in the same
space, as shown in D. E represents the result when the two
tones are produced simultaneously, and illustrates in a strik-
ing manner the effect of interference. F represents the result
when the vowel e is sung at the pitch of C ; and G, when

Fig. 185.

the vowel o is sung on the same key. These are called mano-
metric flames.

188. The phonautograph or phonograph. — Soiind-waves,
however complex, may be caused permanently to record the
succession and variation of their impulses, and thus, as it
were, to inscribe their own autograph. Fig. 185 represents
the original Edison phonograph.

A metallic cylinder A is rotated by means of a crank. On
the surface of the cylinder is cut a shalloAV helical groove

232 MOLAR DYNAMICS.

running around the cylinder from end to end, like the thread
of a screw. A small metallic point, or style, projecting from
the under side of a thin metallic disk D (Fig. 186) which

closes one orifice of the mouth-
piece B, stands directly over the
thread. By a simple device the
cylinder, when the crank is turned,
is made to advance just rapidly
•^^^- ^^^- enough to allow the groove to

keep constantly under the style. The cylinder is covered
with tinfoil. The cone F is usually applied to the mouth-
piece to concentrate the sound-waves upon the disk D.

Kow, when a person directs his voice toward the mouth-
piece, the aerial waves cause the disk D to participate in
every motion made by the particles of air as they beat against
it, and the motion of the disk is communicated by the style
to the tinfoil, producing thereon impressions or indentations
as it passes on the rotating cylinder. The result is that
there is left upon the foil an exact representation of every
movement made by the style. Some of the indentations are
quite perceptible to the naked eye, while others are visible
only with the aid of a microscope of high power. Fig. 187
represents a piece of the foil as it would
appear inverted after the indentations (here ^x^^^W^^;^^

greatly exaggerated) have been imprinted ^

upon it.

The words addressed to the phonograph having been thus
impressed .upon the foil, the mouth-piece and style are tem-
porarily removed, while the cylinder is brought back to the
position it had when the talking began, and then the mouth-
piece is replaced. Now, evidently, if the crank be turned in
the same direction as before, the style, resting upon the foil
beneath, will be made to play up and down as it passes over
ridges and sinks into depressions ; this will cause the disk D

SYNTHESIS OF SOUND-WAVES. 233

to reproduce the same vibratory movements that caused the
ridges and depressions in the foiL The vibrations of the
disk are communicated to the air, and through the air to the
ear ; thus the words spoken to the apparatus may be, as it
were, shaken out into the air again at any subsequent time,
even centuries after, accompanied by the exact accents, into-
nations, and quality of sound of the original

Subsequently Edison improved this instrument by replacing the
metallic foil by a cylinder of hard wax composition, rotating it by
an electric motor, and providing an improved form of style which
engraves upon the wax the most delicate variations of vibratory
motions, and thus, as it were, reproduces speech and musical notes
with all their delicate shades of expression and modulation. In its
improved form it has become a commercial instrument, and is used
in some cases in the place of stenography, the correspondence being
dictated to the instrument and then rej)roduced by means of a
typewriter.

189. Synthesis of sound-waves. — The sound of a tuning-
fork when its fundamental is reenforced by a suitable reso-
nance-cavity, is very nearly a simple tone.

If two nioiuited forks forming the interval of an octave be
sounded together, the tones proceeding from the separate
forks soon blend together into one sound, to which we assign
the pitch of the lower fork, and a quality richer than that of
either. So strong is the illusion, that we cannot believe the
higher fork to be sounding, until we ascertain that placing a
finger on its prongs so as to damp its vibrations at once
changes the timbre, reducing it to the dull, uninteresting
quality of a simple tone. If to these two forks there be
added a fork whose interval is a fifth above the higher of
the first two (i.e. one which gives the second harmonic of the
first), the three tones blend as perfectly as did those of the two
forks ; the only difference perceptible being an additional
increase of richness.

234 MOLAR DYNAMICS.

By sounding simultaneously several forks of different but
appropriate pitch, and with the requisite relative intensities,
Helmholtz succeeded in producing sounds peculiar to various
musical instruments, and even in imitating most of the vowel
sounds of the human voice.

Thus it appears that he has been able to determine, both
analytically and synthetically, that the quality of a given sound
depends upon what overtones combine with its fundamental tone,
and upon their relative intensities ; or, more briefly, upon the
form of the sound-wave, since the form must be determined
by the character of its components.

Section XI.

MUSICAL INSTRUMENTS.

190. Classification of musical instruments. — Musical in-
struments may be grouped into three classes : (1) stringed
instruments ; (2) wind instruments, in which the sound is
due to the vibration of columns of air confined in tubes ; (3)
instruments in which the vibrator is a membrane or plate.
The first class has received its share of attention ; the other
two merit a little further consideration.

191. Wind instrume7its.

Experiment 1. — Fig. 188 represents a set of Quincke's whistles. The
tubes are of the same size, but of varying length. Blow through the
small tube across the lips of the large tube of each whistle in the order of
their lengths, commencing with the longest.

Repeat the experiment, closing the end of the whistle farthest from
you with a finger, so as to make what is called a " closed pipe." i

The pitch of vibrating air-columns, as well as of strings,

1 The eif ect of interference is well shown with these tubes by blowing at the same
time through two tubes of nearly the same length. A peculiar rattling sound is the
result. A very different result is obtained when two tubes a fif'th anart are sounded
together.

CLASSIFICATION OF MUSICAL INSTRUMENTS. 235

varies with the length, and (1) in both stopped'^ and open'^
pipes the number of vibrations is inversely proportional to the
length of the pipe. (2) An open pipe gives a note an octave
higher than a closed pipe of the same length.

Fig. 188.

Experiment 2. — Take some of the longer whistles, and blow as before,
gradually increasing the force of the current. It will be found that only
the gentle current will give the full musical fundamental tone of the tube,
— a little stronger current producing a mere rustling sound ; but when
the force of the current reaches a certain limit, an overtone will break
forth ; and, on increasing still further the power of the current, a still
higher overtone may be reached.

1 The terms " stopped " and " open " apply to only one end of the pipe ; the other,
in both kinds, is always open.

236

MOLAR DYNAMICS.

Fig. 189 represents an open organ-pipe provided with a
glass window A in one of its sides. A wire hoop B has
stretched over it a membrane, and the whole is suspended by
a thread within the pipe. If the membrane be
placed near the upper end, a buzzing sound pro-
ceeds from the membrane when the fundamental
tone of the pipe is sounded; and sand placed on
the membrane will dance up and down in a lively
manner. On lowering the membrane, the buzzing
sound becomes fainter, till, at the middle of the
tube, it ceases entirely, and the sand becomes
quiet. Lowering the membrane still further, the
sound and dancing recommence, and increase as
the lower end is approached.

(3) When the fundamental tone of an open pipe
is p)roduced, its air-column divides into two equal
vibrating sections, with the anti-nodes at the extrem-
ities of the tube, and a node in the middle.

If the pipe be stopped, there is a node at the
stopped end ; if it be open, there is an anti-node
at the open end; and in both cases there is an
anti-node at the end where the wind enters, which
is always to a certain extent open.
A, B, and C of Fig. 190 show respectively the positions of
the nodes and anti-nodes for the fundamental tone and first
and second overtones of a closed pipe ; and A', B', and C
show the positions of the same in an open pipe of the same
length. The distance between the dotted lines shows the
relative amplitudes of the vibrations of the air-particles at
various points along the tube. Now the distance between a
node and the nearest anti-node is a quarter of a wave-length.
Comparing, then, A and A', it will be seen that the wave-
length of the fundamental of the closed pipe must be tvv^ice
the wave-length of the fundamental of the open pipe ; hence

Fig. 189.

SOUNDING PLATES, ETC.

237

the vibration period of the latter is half that of the former;
consequently the fundamental of the open pipe must be an
octave higher than that of the closed pipe.

In the three cases (A, B, and C) of the closed tube, the
length of the air-column is divided into i, |, and | segments
respectively ; hence the corresponding vibration-numbers are

3/2

1 /

\ 1

\ /
\ /

c'^

1 /
\ /

1 \
1 \

\ 1
\ /

/ \

1 \
1 \

\ 1

/ ,
' 1

Fig. 190.

as 1 : 3 : 5, etc. Hence, (4) m a closed tube, only those over-
tones tvhose vibration-numbers correspond to the odd multiples
of that of the fundamental are present.

In the cases of the open tube, the length of the air-column
is divided into |, |, and | segments respectively ; their vibra-
tion-numbers are therefore as 1 : 2 : 3, etc. Hence, (5) in an
open tube, the complete series of overtones corresponding to its
fundamental may be present.

192. Sounding plates, etc.

Experiment ^.—Fasten with a screw the elastic brass plate A (Fig.
191) on the upright support. Strew writing-sand over the plate, draw a

238

MOLAR DYNAMICS.

rosined bass bow steadily and firmly over one of its edges near a corner,
and at the same time touch the middle of one of its edges with the tip
of the finger ; a musical sound will be produced, and the sand will dance
up and down, and quickly collect (1) in two rows, extending across the
plate at right angles to each other. Draw the bow across the middle of

Fig. 191.

an edge, and touch with a finger one of its corners ; the sand will arrange
itself in two diagonal rows (2) across the plate, and the pitch of the note
will be a fifth higher. Touch, with the nails of the thumb and fore-
finger, two points a and 6 (3) on one edge, and draw the bow across the
middle c of the opposite edge, and you will obtain additional rows and a
shriller note.

By varying the position of the point touched and bowed,
a great variety of patterns can be obtained, some of which
are represented in Fig. 192. It will be seen that the effect
of touching the plate with a finger is to prevent vibration at
that point, and consequently a node is there produced. The
whole plate then divides itself up into segments with nodal
division lines in conformity with the node just formed. The
sand rolls away from those parts which are alternately thrown
into crests and troughs, to the parts that are at rest.

INTERFERENCE.

239

193. Interference.

Experiment 4. — C (Fig. 191) is a tin tube made in two parts that
telescope one within the other. The extremity of one of the parts ter-
minates in two slightly smaller branches. Bow the plate, as in experi-
ment 3 (1), place the two orifices of the branches over the segments
marked with the + signs, and regulate the length of the tube so as to
reenforce the note given by the plate, and set the plate in vibration.

/"^^■^^'■'■X^\\

^^^

1 J r

J i L

Fig. 192.

Now turn the tube around, so that one orifice may be over a + segment,
and the other over a — segment ; the sound due to resonance entirely
ceases. It thus appears that the two segments marked + pass through
the same phases together ; likewise the phases of — segments correspond
with one another ; i. e. when one + segment is bent upward, the other
is bent upward, and at the same time the two — segments are bent
dovraward ; for, when the two orifices of the tube are placed over two
+ segments or two — segments, two condensations followed by two rare-
factions pass up these branches and unite at their junction to produce a
loud sound ; but when one of the orifices is over a + segment, and the

240

MOLxiR DYNAMICS.

Fig. 193.

other over a — segment, a condensation passes up one branch at the
same time that a rarefaction passes up the other, and the two destroy
each other when they come together; i.e. the two sound-waves combine
to produce silence.

194. Bells. — A bell or goblet is Subject to the same laws
of vibration as a plate.

Experiment 5. — Nearly fill a large goblet with
water, strew upon the surface lycopodium powder,
and draw a rosined bow gently across the edge of
the glass. The surface of the water will become
rippled with wavelets (Fig. 193) radiating from four
points 90° apart, corresponding to the centers of
four venters into which the goblet is divided, and
the powder will collect in lines proceeding from the
nodal points of the bell. By touching the proper
points of a bell or glass with a finger-nail, it may be
made to divide itself, like a plate, into 6, 8, 10, etc.

(always an even number), vibrating parts.

Experiment 6. — Remove the brass plate (Fig. 191) from its support,

and fasten the bell B (Fig. 194) on the support. Bow the edge of the

bell at some point, and hold the open

tube C in a horizontal position with the

center of one of its openings near that

point of the edge of the bell which is

opposite the point bowed. The tube

loudly reenforces the sound of the bell.

Move the tube around the edge of the

bell and find its nodes.

Thrust the plunger I) into the open

end E of the tube, and find what part of

the length of an open tube a closed tube should be to reenforce a sound

of a given pitch.

195. Vocal organs. — It is difficult to say wliich is more to
be admired, the wonderful capability of the human voice or
the extreme simplicity of the means by which it is produced.
The organ of the voice is a reed instrument situated at the
top of the windpipe, or trachea. A pair of elastic bands, a a
(Fig. 195), called the vocal chords, is stretched across the

^^^^

Fig. 194.

VOCAL ORGANS. 241

top of the windpipe. The air-passage b between these chords
is freely open while a person is breathing ; but when he
speaks or sings, they are brought together so as to form a nar-
row, slit-like opening, thus making a
sort of double reed, which vibrates
when air is forced from the lungs
through the narrow passage, some-
what like the little tongue of a toy
trumpet. The sounds are grave or
high according to the tension of the
chords, which is regulated by muscu-
lar action. The cavities of the mouth
and the nasal passages form a com-
pound resonance - tube. This tube

■^ _ JblG. 195.

adapts itself, by its varying width and

length, to the pitch of the note produced by the vocal chords.
Place a finger on the protuberance of the throat called
"Adam's apple," and sing a low note; then sing a high note,
and you will observe that the protuberance rises in the latter
case, thus shortening the distance between the vocal chords
and the lips. Set a tuning-fork in vibration, open the mouth
as if about to sing the corresponding note, place the fork in
front of it, and the cavity of the mouth will resound to the
note of the fork, but will cease to do so when the mouth
adapts itself to the production of some other note. The
different qualities of the different vowel sounds are produced
by the varying forms of the resonating mouth-cavity, the
pitch of the fundamental tones given by the vocal chords
remaining the same. This constitutes articulation.

Strictly speaking, consonants are not distinct sounds or the rep-

- resentatives of sound. They represent rather, when they precede

vowels, the different positions of the organs of speech from which

(like spring-boards, as it were) the vowels are attacked. i Following

1 A commoii fault with young singers is attempting to " sing the consonants."

242

MOLAR DYNAMICS.

vowels, they represent the position of the organs of speech at the
interruption of the vowel sounds, and the consequent modifications
of these sounds. Consonants are accordingly classified into labials,
dentals, gutturals, etc. The more care exercised in placing the
organs in suitable positions for attack or interruption and the less
sound emitted from these points at the moment of attack, the
clearer is the articulation. Modulations of the voice in conversa-
tion take place usually in musical intervals. Singing differs from
speaking chiefly in the manner in which the vocal sounds are modi-
fied. In both, the sustained sounds are vowel sounds. In fact
only vowel sounds are musical, and any language is musical in
proportion to the number of vowels it contains. Thus Greek is a
more musical language than Latin, and Italian than German.

196. The ear. — In Fig. 196, A represents the external ear-

FlG. 196.

passage; (x is a membrane, called the tympanum, stretched
across the bottom of the passage, and thus closing the orifice
of a cavity b, called the drum; c is a chain of small bones
stretching across the drum, and connecting the tympanum

THE EAR. 24B

with the thin membranous wall of the vestibule e ; ff are a
series of semicircular canals opening into the vestibule ; g is
an opening into another canal in the form of a snail-shell g\
hence called the cochlea (this is drawn on a reduced scale) ;
c? is a tube (the Eustachian tube) connecting the drum with
the throat; and h is the auditory nerve. The vestibule and
all the canals opening into it are filled with a transparent
liquid. The drum of the ear contains air, and the Eustachian
tube forms a means of ingress and egress for air through the
throat.

Now how does the ear hear ? and how is it able to dis-
tinguish between the infinite variety of form, rapidity, and
intensity of aerial sound-waves so as to interpret correctly
the corresponding quality, pitch, and loudness of sound ?
Sound-waves enter the external ear-passage A as ocean-waves
enter the bays of the sea-coast, are reflected inward, and
strike the tympanum. The air-particles, beating against this
drum-head, impress upon it the precise wave-form that is
transmitted to it through the air from the sounding body.
The motion received by the drum-head is transmitted by the
chain of bones to the membranous wall of the vestibule.
From the walls of the spiral passage of the cochlea project
into its liquid contents thousands of fine elastic threads or
fibres, called "rods of Corti." As the passage becomes smaller
and smaller, these vibratile rods become of gradually dimin-
ishing length and size (such as the wires of a piano may
roughly represent), and are therefore suited to respond sym-
pathetically to a great variety of vibration-periods. This
arrangement is sometimes likened to a " harp of three thou-
sand strings " (this being about the number of rods). The
auditory nerve at this extremity is divided into a large num-
ber of filaments, like a cord unraveled at its end, and one
of these filaments is attached to each rod. Now, as the
sound-waves reach the membranous wall of the vestibule,

244 MOLAR DYNAMICS.

they set it, and by means of it the liquid contents, into fo7xed
vibration, and so through the liquid all the fibers receive an
impulse. Those rods whose vibration periods correspond
with the periods of the constituents forming the compound
wave are thrown into sympathetic vibration. The rods stir
the nerve filaments, and the nerve transmits to the brain the
impressions received. Much as a piano when its dampers are
raised and a person sings into it, may be said to analyze each
sound-wave, and show by the vibrating strings of how many
tones it is composed, as well as their respective pitches, and by
the amplitude of their vibrations their respective intensities ;
so, it is thought, this wonderful harp of the ear analyzes
every complex sound-wave into a series of simple waves.
Tidings of the disturbances are communicated to the brain,
and there, in some mysterious manner, these disturbances are
interpreted as sound of definite quality, pitch, and intensity.

PART 11.

MOLECULAR DYNAMICS. -HEAT.

Section I.

THEORY OF HEAT.

In the preceding pages the theory of heat has been several
times anticipated ; we are now better qualified to judge of its
validity.

197. Energy of mass motion convertible into heat.

Experiment 1. — Hold some small steel tool upon a rapidly revolving
dry grindstone; a shower of sparks flies from the stone. Place a ten-
penny nail upon a stone and hammer it briskly ; it soon becomes too hot
to be handled with comfort, and we may conceive that if the blows were
rapid and heavy enough, it might soon become red hot. Eub a desk
with your fist, and your coat-sleeve with a metallic button ; both the
rubbers and the things rubbed become heated.

Yon observe that in every case heat is generated at the
expense of work or mass energy ; i.e. mass energy destroyed
becomes heat. When the brakes are applied to the wheels of
a rapidly moving railroad train, its energy is converted into
heat, much of which may be found in the wheels, brake-
blocks, and rails. The meteorites, or " shooting-stars," which
are seen at night passing through the upper air, sometimes
strike the earth, and are found to be stones heated to a light-
giving state. They become heated when they reach our
atmosphere, in consequence of their motion being checked by
the resistance of the air.

246

MOLECULAR DYNAMICS.

198. Heat convertible into mass energy.

Experiment 2. — Take a tliin glass flask A (Fig. 197), half fill it witli
water, and fit a cork air-tight into its neck. Perforate the cork, insert a
glass tube . bent as indicated in the figure, and extend it into the water.
Apply heat to the flask; soon the liquid rises in the
Q u u tube, and flows from its upper end.

Here heat produces mechanical motion, and
does work in raising a mass in opposition to
gravitation. Every steam engine is a heat en-
gine, i.e. the power of steam is due to its heat.
The steam which leaves the cylinder of an en-
gine, after it has set the piston in motion, is
cooler than when it entered.

It will be shown hereafter that in all cases
when work is done by heat without waste or
loss, the quantity of heat consumed is propor-
tional to the work done ; and, conversely, by
the performance of a definite quantity of work
an equivalent quantity of heat is produced; in
other words, there is a definite quantitative
relation between heat and work.

If heat be consumed, and mechanical work thereby per-
formed, we are justified in saying that heat has transformed
itself into mass energy ^ and, conversely, if mass energy be
expended and heat thereby produced, we may say that mass
energy has transformed itself into heat.

Now, when the appearance of one thing is so connected
with the disappearance of another that the quantity of the
thing produced can be calculated from the quantity of that
which disappears, we conclude that the one is formed at the
expense of the other, and that they are only different forms
of the same thing. We have, therefore, reaison to believe
that heat is of the same nature as mass energy ; i.e. it is only
another form of energy.

Fig. 197.

THEORY OF HEAT. 247

199. Theory of heat. — A body loses motion in communi-
cating it. The hammer descends and strikes the anvil ; its
motion ceases, but the anvil is not sensibly moved ; the only
observable effect produced is heat. Instead of a motion of
the hammer and anvil, there is now an increased vibratory
motion of the molecules that compose the hammer and anvil,
— simply a change from molar to molecular motion. Of course,
this latter motion is invisible. According to this view, heat
is hut a name for the energy of vibration of the Tnolecules of a
body, or, briefly, heat is molecular kinetic energy. The
science which treats of heat as a form of energy is called
thermodynamfiics.

A body is heated by having the motion of its molecules
quickened, and cooled by parting with some of its molecular
motion. Cold is comparable to rest, heat^ to motion. One
body is hotter than another when the average kinetic energy of
the molecides in it is greater than in the other.

As late as the beginning of tlie present century heat was gen-
erally regarded as " a sensation which the presence of fire" (an
" igneous ^uicZ," "matter of heat," called sometimes " caloric ")
" occasions in animate and inanimate bodies." A text-book of that
period makes this significant statement : " There is fire in the wood,
and there is air in the field, though we do not perceive either while
at rest. Eubbing two pieces of wood does not create fire any more
than the blowing of the wind creates air. Motion renders both
perceptible.'''' The former and the more modern views are in har-
mony in attributing the immediate cause of the sensation to motion.
According to the former view, the sensation is produced by putting
an imaginary fluid in motion; according to the modern view it is
produced by quickening the motion of the molecules of a body.

The material theory became untenable when it was shown by
Count Eumford ^ that the quantity of heat that may be evolved by

1 Whenever there is occasion to speak of the sensation which heat is capable of
producing, it should never be called heat, but it should be termed a heat sensation or
the sensation of heat.

2 The great discovery of the non-materiality of heat was made by an American,
Benjamin Thompson (Count Eumford), then (1798) residing in Munich. This dis-

248 MOLECULAR DYNAMICS.

friction, as, for instance, in the boring of cannon, is practically
limitless, or is limited only by the mechanical power available.
Now according to this theory, when a piece of metal is rubbed the
caloric is rubbed or squeezed out of it ; but, as Rumford argued,
" anything which a body can continue to furnish without limitation
cannot possibly be a material substance." At about the same time
Davy showed that two pieces of ice may be melted by rubbing them
together in a space whose temperature is below the melting point.

200. Heat, the lowest for m of energy. — Heat is often spoken
of as the '' lowest form of energy." That is, all other forms
of kinetic energy tend to transform themselves into the
"lower" form of heat ; as water tends to seek a lower level.
When energy is spent in doing work, that portion which
appears in no other form appears as heat.

Section II.

SOURCES OF HEAT.

201. Mechanical eiiergy a source of heat. — As heat is
energy, so all heat must originate in some form of energy,
i.e. by the transformation of some other form of energy into
heat.

In the preceding section it was shown that heat may be
generated at the expense of molar motion, i.e. molar motion
checked usually results in molecular motion, the energy of
which is heat. By friction, by compression, by percussion,
or by any process by which mass motion is arrested, heat is
mechanically generated.

202. Chemical union a source of heat.

Experiment 1. — Take a glass test-tube half full of cold water and pour
into it one-fourth its volume of strong sulphuric acid. The liquid almost
instantly becomes so hot that the tube cannot be held in the hand.

covery lies at tlie foundation of the dynamical theory of heat, and directly led to the
grandest doctrines of modern science, the correlation and the conservation of energy.
(See p.304 )

ANIMAL HEAT AND MUSCULAR MOTION. 249

When water is poured upon quicklime, heat is rapidly
developed. The invisible oxygen of the air combines with
the constituents of the various fuels, such as wood, coal, oils,
and illuminating-gas, and gives rise to what we call burning,
or co77ihustion, by which a large amount of heat is generated.
In all such cases the heat is generated by the combination or
clashing together of molecules of substances that have an
affinity (i.e. an attraction) for one another. Before union
the energy of the molecules is of the same kind as that of a
stone on a shelf. When the shelf is withdrawn, gravity con-
verts the potential energy of the stone into kinetic energy ;
so affinity converts the potential energy of the molecules into
kinetic energy of vibration, i.e. into heat.

When a definite mass, say of carbon or hydrogen, is burned,
the quantity of heat produced is definite ; hence the different
fuels have a definite heat value, which depends in part upon
the combustion equivalents of their constituents.

In a majority of cases chemical union is attended by the evolu-
tion of heat ; but in some cases work has to be done by heat upon
separate elements to force them to combine either directly or indi-
rectly ; hence in such cases the union is attended by a consumption
or disappearance of heat, and the decomposition of compounds thus
formed is attended by an evolution of heat.

203. Origin of animal heat and muscular motion. — The
plant finds its food in the air (principally the carbonic acid
in the air) and in the earth, in a condition analogous to that
of a fallen weight ; but, by the agency of the sun's radiation,
work is performed upon this matter during the growth of the
plant ; potential energy is stored in the plant, — the weight
is drawn up as it were. The animal now finds its food in the
plant, appropriates the potential energy stored in the plant,
and, by chemical action, chiefly by the union of carbon and
hydrogen with oxygen, this energy is converted into the en-
ergy of motion in the form of heat and muscular motion, —

250 MOLECULAR DYNAMICS.

the weight falls and its energy becomes kinetic. The plant,
then, may be regarded as a machine for converting energy
of motion received from the sun into potential energy ; the
animal, as a machine for transforming it again into energy
of motion.

204. The sun as a source of energy. — The sun is not only
the source of the energy exhibited in the growth of plants, as
well as of the muscular and heat energy of the animal, but is
also the source, directly or indirectly, of very nearly all the
energy employed by man in doing work. Our coal-beds, the
results of the deposit of vegetable matter, are vast store-
houses of the sun's energy, rendered potential during the
growth of the plants many ages ago. Every drop of water
that falls to the earth and rolls its way to the sea, contribut-
ing its mite to the immense water-power of the earth, and
every wind that blows, derives its power directly from the
sun.

205. Dissipatio7i of energy.

Work is done by heat only when it passes from a higher to a
lower level, i.e. from a higher temperature to a lower temperature.
In other words heat would not be available for doing work if all
matter were reduced to the same temperature. Heat has a ten-
dency to become uniformly diffused. The temperature of the entire
material universe tends to uniformity. This does not imply that
the quantity of energy in the universe changes, but it does imply
that the quantity of energy available to man for doing work is
diminishing. A railroad car with its furniture and all objects on
board the car may be considered as constituting a system of bodies.
If the car be in rapid motion, it possesses a large quantity of energy
and each body in the system possesses energy proportionate to its
mass, but the energy of no one of these bodies is available for doing
work upon another. Why ? Because all have like velocities, and
no one can impart velocity to another. So in a system where all
the molecules have a like velocity, there can be no transfer of
velocity or energy, and hence no work can be done within the
system by any of its members.

TEMPERATURE DEFINED. 251

Now since all forms of energy tend toward this lower form of
energy, and the temperature of all matter tends towards a uni-
formity, and since in proportion as uniformity of temperature is
reached the quantity of available energy in the universe is dimin-
ished, we are forced to the conclusion expressed by Tait that " The
available energy of the universe tends to zero." ^ This. is called by
Lord Kelvin the Doctrine of Dissipation of Energy.

Section III.

TEMPERATURE.

206. Temjperatuve defined. — The words ivarm, hot, cool,
cold are associated in our minds with a series of sensations
which we suppose to indicate a corresponding series of states
of an object with respect to heat; that is, the agent which
produces these sensations is heat. These are all temperature
terms, and refer to the state of an object with reference to
heat. Tertijiievature is the state of matter in respect to heat.
When the quantity of heat in a body increases, its tempera-
ture is said to rise ; and when this diminishes, its temperature
is said to fall. The relation which temperature bears to heat
is analogous to that which hydrostatic pressure bears to
water. Water flows from high level to low level ; heat flows
from high temperature to low temperature. When we pour
water into a vessel, the level rises ; so heat entering a body
raises its temperature (unless it is transformed in doing work).
It takes more water to fill a large vessel to a given depth
than a small one ; it takes more heat to raise the temperature
of a body of large mass a certain amount than to raise the
temperature of a smaller mass of the same substance an equal

1 "In the beginning " poiaits to a far distant period when all the energy of the
physical universe was in the available form. Physical science foresees a time when
all available energy shall become zero and all the processes of nature must cease.
" The marvellous mechanism of nature will then have run down, and no further
motion or life-process will be possible unless some new order intervenes of which
we have no knowledge or conception."

252 MOLECULAR DYNAMICS.

amount. Depth of water in vessels of varying size is not
proportional to the quantity of water they contain ; tempera-
ture of bodies of varying mass is not necessarily proportional
to the quantity of heat they contain. For example, a pint of
water at a given temperature does not contain the same quan-
tity of heat as a gallon of water at the same temperature.

If body A be brought in contact with body B, and A tend
to impart heat to B, then A is said to have a higher tempera-
ture than B. Temperature is sometimes spoken of as the state
of a body ivith reference to its tendency to commuiiicate heat to,
or receive heat from, other bodies. The direction of the flow
of heat determines which of two bodies has the higher tem-
perature. If the temperature of neither body rise at the
expense of the other, then both have the same temperature,
or are said to be in thermal eqidlibrium.

Temperature depends on the average kinetic energy of the
molecules. The temperature of a substance increases propor-
tionally to the mean square of the velocity of vibration of its
molecules. Bodies have the same temperature when the
average energy of the molecules of each is the same.

207. Temperature a relative term.

As the term is now used, temperature is a relative expression,
not an expression of absolute quantity. It is regarded rather as a
quality capable of greater or less intensity, than as a quantity which
may be added to or subtracted from other quantities of the same
kind.

For instance, if the temperatures of two bodies be respectively
25° C. and 35° C, we say with truth that their temperatures differ
by 10 centigrade degrees ; but we cannot say with proprietj^ that a
temperature of 25° subtracted from a temperature of 35° leaves a
temperature of 10°, or that the two temperatures added together
give a temperature of 60°. ^

There is no propriety in the expression "twice as hot." The
term temperature is similar to the term hardness, insomuch as we
are able to construct a scale for both, so that a body may have a
definite place in the scale and be less hot or less hard than some-

USE OF A THERMOMETER. 253

thing above it, and yet we are not able to estimate either tempera-
ture or hardness quantitively.

Section IV.

THERMOMETRY.

208. Use of a thermometer. — A thermometer is an instru-
ment for indicating temperature, i.e. tlie difference between
the temperature of a given body and some standard tempera-
ture (§ 211). For the difference between two standard tem-
peratures, such as the melting point of ice and the boiling
point of water, is one capable of accurate subdivision into
any number of equal parts, which form successive equal steps
from the lower to the higher temperature.

209. TeTuperature indicated by expansion. — The effects of
expansion by heat are well illustrated in the common ther-
mometer. As its temperature rises, both the glass and the
mercury expand ; but, as liquids in general are more expansi-
ble than solids, the mercury expands much more rapidly than
the glass, and the apparent expansion of the mercury, shoivn hy
its rise in the tube, is the difference between the actual increase of
volume of the mercury and that of the capacity of the glass ves-
sel containing it. The thermometer, then, primarily indicates
changes of volume ; but as changes of volume in this case are
caused by changes of temperature, it is commonly used for
the more important purpose of indicating temperature.

If a thermometer be brought into intimate contact with a
body whose temperature is sought, as, for instance, a liquid
into which it is plunged, or the air in a room, the mercury in
the tube rises or falls until it reaches a certain point, at
which it remains stationary. We then know that it is in
thermal equilibrium with the surrounding body. Hence the
reading, as it is called, of the thermometer indicates not only
the temperature of the mercury, but of the surrounding body.

254 MOLECULAR DYNAMICS.

210. Construction of a thermometer. — A thermometer con-
sists generally of a glass tube of capillary ^ bore, terminating
at one end in a bulb. The bulb and part of the tube are filled
with, mercury, and the space in the tube above the mercury
is a partial vacuum. On the tube, or on a plate of metal
behind the tube, is a scale to show the hight of the mercurial
column.

211. Standard temjper attires. — That a thermometer may
indicate any definite temperature, it is necessary that its
scale should relate to some definite and unchangeable points
of temperature. Fortunately Nature furnishes us with two
convenient standards. It is found that under ordinary atmos-
pheric pressure ice always melts at the same temperature,
called the melting iDoint. Again, the temperature of steam
rising from boiling water under the same pressure is always
the same.

212. Graduation of thermometers. — First, the bulb of a
thermometer is placed in melting ice (Fig. 198) and allowed
to stand until the surface of fhe mercury becomes stationary,
and a mark is made upon the stem at that point, which indi-
cates the melting point. Then the instrument is suspended
in steam rising from boiling water (Fig. 199), so that all but
the very top of the column is in the steam.

The bulb is placed in a metallic vessel, M, with a narrower
upper part, A. This narrower part is surrounded by a larger
part, B. By observing the arrows it is seen that steam sur-
rounds the inner part, and thus prevents its cooling ; it
escapes by the tube D. The orifice of D is large enough to
allow the steam to escape freely, and thus prevent a pressure
inside the vessel greater than the atmospheric pressure. To
guard against such a contingency a pressure gauge m is in-
serted ill the vessel. The liquid in both arms of the gauge

1 In consequence of refraction of light at the cylindrical surface of the glass tube,
the diameter of the cylinder of mercury in the bore appears magnified.

GRADUATION OF THERMOMETERS.

255

must be kept at the same level througliout the operation.
The mercury rises in the stem of the thermometer until its
temperature becomes the same as that of the steam, when it
becomes stationary. A barometer is consulted, and due allow-
ance having been made for atmospheric pressure at the time,
a mark is placed on the stem to indicate the boiling point.
This boiling point is the temperature of steam at a pressure

Fig. 198.

Fig. 199.

of 760 mm of mercury at 0° C. in the latitude of Paris (48°
50'), 60 meters above sea-level. Then the space between the
two points found is divided into a convenient number of
equal parts (provided that the bore of the tube is of uniform
diameter) called degrees, and the scale is extended above and
below these points as far as is desirable.

Two methods of division are adopted in this country (see a
and h, Fig. 200) : by one, the space is divided into 180 equal
parts, and the result is called the Fahrenheit scale, from the
name of its designer ; by the other, the space is divided into

256

MOLECULAR DYNAMICS.

100'

0°

■17.8°

212^

100 equal parts, and the resulting scale is called centigrade,
-which, means one hundred steps. In the Fahrenheit scale,
which is generally employed for ordinary household purposes,
^ the melting and boiling points are marked

a respectively 32° and 212°. The of this

scale (32° below melting point), which is
about the lowest temperature that can be ob-
tained by a mixture of snow and salt, was
incorrectly supposed to be the lowest temper-
ature attainable. The centigrade scale, which
is generally employed by scientists, has its
melting and boiling points more conveniently
marked, respectively 0° and 100°. A temper-
ature below 0° in either scale is indicated
by a minus sign before the number. Thus,
- 12° F. indicates 12° below 0° (or 44° below
melting point), according to the Fahrenheit
scale. Under F. and C. (in the left column.
Fig. 201), the two scales are placed side by
side, so as to exhibit at intervals a compara-
tive view. The Fahrenheit and centigrade
scales agree at — 40°, but diverge both ways
from this point.

213. Conversion from one scale to the other.
— Since 100° C. =c= 180° F., 5° C. =^ 9° F.,
or 1° C. =crr I of 1° F. Hence, to convert
Fig 200 centigrade degrees into Fahrenheit degrees,

we multiply the number by f ; and to convert
Fahrenheit degrees into centigrade degrees we multiply by |.
In finding the temperature on one scale that corresponds to a
given temperature on the other scale, it must be remembered
that the number that expresses the temperature on a Fahren-
heit scale does not, as it does on a centigrade scale, express
the number of degrees above melting point. For example.

0°

DEVELOPMENT OF THE THERMOMETER,

257

52° on a Fahrenheit scale is not 52° above melting point, but
52° - 32° = 20° above it.

Hence, to reduce a Fahrenheit reading to a centigrade read-
ing, first siiMract 32 from the given number, and then multiply
bij |. Thus,

|(F-32)-:C.

To change a centigrade
reading to a Farhenheit
reading, first multirply the
given number by f , and then
add 32. Thus.

|C + 32=:F.

214. Development of the
thermometer.

Tin melts 233=

Water boils 100°
Alcohol boils 78°

Ether boils 35°
Ice melts 0°

Though the invention of Mercury .freeze^8.8° H-37.9° 234.2° H

the thermometer has been
ascribed to various scien-
tific men, it did not assume

a practical shape until 1620, Aicohoifreezes-j 30.50 H-202.9° h.-.s^h

at the hands of Drebel, a
Dutch physician. Halley

mihstitntPfl mprrnrv for Lowest temperature yet attained
SUDSUlUieCl merCUiy lOr estimated to be about-2200

spirit in 1730 ; Reaumur

of Paris modified the instru- ~2^^°

ment in 1730, and Fahren- ^^^- ^oi-

heit of Danzig in 1749 ; Celsius of Upsala, Sweden, improved it in

1742 by adding the scale now known as centigrade. 1

451°

212°
172.4°

^2 ir^ "<

506° H 878.4°

3730
351°

308°
273°

C59'.4<5

522.4°"
459.4°

215. Self registering, or maximum and mirdmum thermorp.c-

ters.

These are thermometers which enable us to ascertain the highest
or lowest temperature to which they have been exposed in a given

1 The Reaumur thermometer is used generally in Germany, the centigrade in
France, and the Fahrenheit in England, "A prophet is not without honor save in
his own country."

258

MOLECULAR DYNAMICS.

interval of time, ordinarily a day. The maximum thermometer
(the lower one, Fig. 202) is an ordinary mercurial thermometer,
except that the bore of the tube near the bulb is reduced in such a
way that while the expansion of the mercury is sufficient to force
the liquid past the constriction, the cohesion of the liquid is insuffi-
cient to draw it back again when the temperature falls. To set the
thermometer it is placed in a vertical position and shaken. This
causes the mercury to return past the constriction to the bulb, and
then the instrument indicates the same temperature as that of the
air. The instrument is then placed in a horizontal position as shown
in the figure.

The 7ninimum thermometer (the upper one. Fig. 202) contains
alcohol, since the freezing point of alcohol is far below that of
mercury. The bore of the tube contains a little index of metal or
black glass which moves with a little friction in the tube. It is
entirely enveloped in the spirit, and the action is as follows : The

Fig. 202.

instrument is set by placing it first in an inverted position to allow
the index to run down the tube to the end of the liquid column,
and it is then placed in a horizontal position. If the temperature
rise, the spirit flows past the index without disturbing it. If the
temperature fall below that at which the instrument was set, the
capillary action between the spirit and the index is such as to pre-
clude its leaving the index ; accordingly this is drawn back with the
spirit, its upper end being always flush with the extremity of the
liquid column, and ultimately marking the lowest temperature
reached by the column.

216.

Requirements of a thermometric substance.

The sui)stances most commonly employed for thermometric pur-
poses are mercury, alcohol, and air. The requirements are as

STANDAED THERMOMETER. 259

follows : (1) The substance should be uniformly expansible. In this
respect the air-thermometer most nearly meets the requirement ;
but there are great inconveniences attending its use. The expansion
of mercury is nearly uniform between the melting and boiling points
on the scale and for a considerable range on each side of them.
Alcohol, on account of lack of uniformity in its rate of expansion,
is ill-adapted for this use. (2) A thermometric liquid should have a
high boiling point. A mercury thermometer will indicate tempera-
ture as high as about 327° C. (620° F.). The boiling point of alcohol
is below that of water. (3) The freezing point of the liquid should
be low. This furnishes the only reason for the use of alcohol. (4)
The sensibility of the liquid to sudden changes of temperature is
of importance. On account of the high conductivity of mercury
and its low specific heat (§ 221), a mercurial thermometer acquires
thermal equilibrium with the surrounding body more quickly than
any other liquid thermometer.

217. Standard thermoTneter.

As its name indicates, this instrument is used as a standard for
reference, and for testing from time to time the accuracy of ther-
mometers used for ordinary observations.

It is a thermometer which has been compared for every degree
with an air thermometer, and has a table of corrections accom-
panied with a certificate of authority from the laboratory where it
was compared. Owing to the fact that glass after having been
fused does not immediately return to its normal density, it is neces-
sary to use for a standard thermometer a tube which has lain
several years after being filled. The contraction of the bulb causes
a thermometer to read too high if it was graduated before the
contraction was completed. The result of this defect in a ther-
mometer is called the displacement of zero. The graduation of a
standard thermometer is made on the glass stem, and thus com-
plication due to difference of dilation of different substances is
avoided.

218. Determination of extremely high temperatures. Pyrom-
eter.

Any contrivance for determining temperature above the range
of a mercurial thermometer (327° C.) is called a pyrometer. There
are several varieties. Some are constructed on the principle that

260

MOLECULAR DYNAMICS.

Fig. 203.

a change of temperature affects the electrical
resistance of a metal. A hydropyrometer indi-
cates high temperatures by the number of
degrees a given mass of water is raised, by
the immersion in it of a platinum ball of
known weight after it has acquired the tem-
perature, for instance, of a furnace or oven
to be tested. DanielPs pyrometer is one of
the most useful in practice. The indications
are obtained from the difference in expan-
B sion of a platinum bar A (Fig. 203) and a
tube of black lead B in which the bar is con-
tained. The index C moves over the scale
D as the metal rod expands. The degrees
on this scale are easily converted into those of
Fahrenheit or centigrade scales.

Section V.

CALORIMETRY.

219. Distinction between the questions '^ hoiv hot " and " how
much heat.^^ — The former, like the question '^how sweet/'
when applied to a solution of sugar, is answered only rela-
tively. The latter, like the question "how much sugar in
the solution," is answered quantitively. Sweetness and tem-
perature are independent of the mass of the body. Quantity
of sugar depends upon the sweetness and the mass of the
liquid ; quantity of heat depends upon the temperature and
the mass of the body. A pint of boiling water is as hot as
a gallon of the same ; but the latter contains eight times as
much heat. Temperature depends on the average kinetic energy
of the molecules. Quantity of heat is the product of the avej^age
kinetic energy of the molecules multiplied by the number of
molecules.

220. Thermal units. — A thermal unit is the quantity of
heat required to produce a definite effect. The thermal unit
generally adopted is the ccdorie, which is the quantity of

HEAT CAPACITY, SPECIFIC HEAT. 261

heat necessary to raise one kilogram of water from 4° to 5°
C.^ The thermal unit in the C. G. S. system is the grain-calorie,
sometimes called the smaller calorie, which is the quantity of
heat required to raise one gram of water from 4° to 5° C. In
defining a thermal unit it is necessary to state the tempera-
tures between which the water is raised, because, although
the quantity of heat required to raise a given quantity of
water one degree is very nearly the same at different tempera-
tures, and in practice is usually regarded as the same, yet the
quantity required is a very little greater at high temperatures
than at low temperatures (see § 223). The operation of meas-
uring heat is called calorimetry.

221. Heat capacity, specific heat. — The expression heat
capacity applied to a body refers to the quantity of heat
necessary to raise the temperature of the body 1°. The
expression specific heat is applied only to some particular
substance and refers to the quantity of heat required to raise
one kilogram of that substance from 4° to 5° C. It is apparent
that the specific heat of a substance is the heat capacity of 1 unit
of mass of that substance.

Experiment 1. — Mix 1 K of water at 0° with 1 K at 20° ; the tempera-
ture of the mixture becomes 10°.. The heat that leaves 1 K of water
when it falls from 20° to 10° is just capable of raising 1 K of water from
0° to 10°.

Experiment 2. — Take (say) 300 g of sheet lead, make a loose roll of it,
and suspend it by a thread in boiling water for about five minutes, that
it may acquire the same temperature (100° C.) as the water. Eemove
the roll from the hot water, and immerse it as quickly as possible in
300 g of water at 0°, and introduce the bulb of a thermometer. Note the
temperature of the water when it ceases to rise, which will be found to

1 Authorities do not agree on the temperattire limits for this unit. Some German
authorities give 15° to 16'' C. Regnault chose 0° to 1° C, and this has been quite
generally adopted in scientific treatises. There seem, however, to be good reasons
for a departure from this custom, and we have chosen the limits proposed by
Glazebrook in his recent treatise on Heat, viz., 4° to 5° C.

262 MOLECULAR DYNAMICS.

be about 3° (accurately 3.3° +). The lead cools very much more than the
water warms. The temperature of lead falls about 33° for every degree
an equal mass of water is warmed.

From the first experiment we infer that a body in cooling
a certain number of degrees gives to surrounding bodies as
much heat as it takes to raise its temperature the same number
of degrees. From the second experiment we learn that the
quantity of heat that raises 1 K of lead from 3.3° + to 100°,
when transferred to water, can raise 1 K of water only from
0° to 3.3°. Hence we conclude that equal quantities of heat,
applied to equal masses of different substances, raise their
temperatures unequally.

If equal masses of mercury, alcohol, and water receive
equal quaDtities of heat, the mercurjrwill rise 30°, and the
alcohol nearly 2°, for every degree the water rises. From
this we infer that to raise equal masses of each of these
substances 1° requires 30 times as much heat for the water
as for the mercury, and twice as much as for the alcohol.
Since a given quantity of heat affects the temperature of a
given mass of water less than that of an equal mass of mer-
cury or alcohol, water is said to have greater specific heat
than these substances. It is also apparent that a given mass
of water in cooling imparts to surrounding bodies more heat
than the same masses of mercury and alcohol would impart
in cooling the same number of degrees, in proportion to its
greater specific heat.

222. Method of measuring specific heat.

There are at least four methods practiced, only one of which, the
"method of mixtures," will be considered. A known mass m (in
kilograms) of the substance of which the specific heat is required is
taken, as in Experiment 2, and heated to a known temperature t-^
(C) ; then it is mixed with (or immersed in) a known mass of water
m2 at a lower temperature ^2, and as soon as thermal equilibrium is

SPECIFIC HEAT. 263

established throughout, the temperature of the mixture t is taken.
Let s represent the specific heat of the substance sought. Then the
quantity of heat lost by the substance is m X s (^i — t) calories ;
while that gained by the water is m^ {t — t^) calories. Now if no
heat be lost during the operation, ?n X s (ti — t) = 1112 [t — ^2), whence

^ ~ _ V • For example, taking the quantities obtained in the

experiment above, we find for lead (300 g=^=.3 K) s = '' )' . r-^

.»j (100 — 0.0)
= .034 calorie.

223. Specific heat of the same substances at different tem-
peratures and in the three states of matter . — The specific heat
of solids and liquids usually increases slightly with the tem-
perature, and diminishes with increase of density. The spe-
cific heat of water at 0°, 40°, and 80° is respectively 1, 1.003,
and 1.0089 calories. Substances in the liquid state usually
have a higher specific heat than in the solid or gaseous state.
Thus, water has nearly double the specific heat of ice, and a
little more than double the specific heat of steam.

The mean specific heat of a substance between 0° and t° is the
average quantity of heat {e.g. of calories) per degree required in
heating a unit mass of the body from 0° to t°. Let h be the total
number of thermal units required to heat the unit mass of the
substance from 0° to ^°, then the mean specific heat s between 0°
and t° is expressed by the formula

' = !■

The specific heat of any perfect gas measured by its mass is
independent of temperature and density ; for an imperfect gas
(vapor), it increases with the temperature, and diminishes with
increase of density. The specific heat of all perfect gases measured
by volume depends on the number of atoms in the molecule, being
proportional to that number. Thus the specific heats of all diatomic
perfect gases are nearly the same ; that of a triatomic gas would be
to these as 3:2, etc.

264 MOLECULAR DYNAMICS.

Reference Tables.
Table of mean specific heat between 0° C. and 100° C.

Copper 095

Sulphur 2026

Glass 1770

Iron ....... .113

Mercury 033

Lead 031

Specific heat of the same substance in different states.

Solid Liquid Gaseous

Water 504 . . . 1.000 . . . .480

Bromine 083 . . . .106 . . . .055

Lead 031 . . . .040

Alcohol 55-.77 ... .45

224. Great ccq^acity of water for heat. — Water requires
more lieat to warm it, and gives out more in cooling through
a given range of temperature, than any other substance except
hydrogen. The quantity of heat that raises a kilogram of
water from 0° to 100° C. would raise a kilogram of iron from
0° to 800° or 900° C, or above a red heat. Conversely, a
kilogram of water in cooling from 100° to 0° C. gives out as
much heat as a kilogram of iron in cooling from about 900°
to 0° C.

"The vast influence which the ocean must exert as a
moderator of climate here suggests itself. The heat of
summer is stored up in the ocean, and slowly given out
during the winter. This is one cause of the absence of
extremes in an island climate."

The high specific heat of water is utilized in heating
buildings by hot water.

225. Relation between specific heat and atomic mass.

The heat energy of a molecule of hydrogen is equal to that of a
molecule of oxygen at the same temperature ; and a mass of
hydrogen contains sixteen times as many molecules as an equal
mass of oxygen. Hence a given mass of hydrogen possesses sixteen

EFFECTS OF HEAT. 265

times as much heat energy as an equal mass of oxygen at the same
temperature. Therefore, to produce a given rise in the temperature
of a mass of hydrogen, sixteen times as much heat is required as
for an equal mass of oxygen ; hence the specific heat or thermal
capacity of hydrogen is sixteen times that of oxygen.

226. Specific heat of elementary gases varies inversely as
their atomic masses ; or, the product of the specific heat and
atomic mass is eonstant.

When the molecules are constrained by cohesion, as in liquids
and solids, a part of the heat applied to a body is spent in raising
its temperature and a part in doing internal work in overcoming
cohesion between the molecules of the body and in forcing them to
take up new positions. The greater the portion of heat consumed,
i.e. converted into potential energy, in doing internal work, the less
there is left to raise its temperature. Hence the law as given for
gases holds only approximately for liquids and solids.

Section VI.

effects of heat. expansion.

Having learned something of the nature of heat and the
methods by which it is measured, w^e will next direct our
attention to some effects it produces, viz. expansion and
change of state. The first, as we have learned, furnishes a
means of measuring temperature and leads to a fuller study
of gases than has yet been made. Under the second effect
we study liquefaction and vaporization.

227. Experiments illustrating expansion of solids, liquids,
and gases.

Experiment 1. — Take two brass tubes, one of a size that will permit
it just to enter the bore of the other. Heat the smaller tube ; it will not
in its expanded state enter the other. Thrust the heated tube into cold
water ; its temperature falls, and it now enters the bore of the other tube. .
"Heat expands," but "cold" does not "contract." Cohesion, when a

266

MOLECULAR DYNAMICS.

diminution of heat (which acts as a repellent force) permits, causes a
solid or liquid body to contract. Cold is a term of negation signifying
merely a greater or less deficiency of heat ; it is not an entity, hence it
cannot be the direct cause of any phenomenon.

Experiment 2. — Fig. 204 represents a thin brass plate and an iron
plate of the same dimensions riveted together so as to form what is called

a compound bar. Place the bar edgewise

i"" ' ""'I in a flame, dividing the flame in halves

(one half on each side of the bar) so
that both metals may be equally heated.
The bar, which was at first straight, is now bent, owing to the unequal
expansion of the two metals on receiving equal increments of temperature.

Fig. 204.

Fig. 205.

Fig. 206.

When heated above the normal temperature, the brass, which is more
expansible, will be on the convex side ; when cooled below the normal
temperature, it will be on the concave side, since it contracts more
rapidly than iron.

Metallic thermometers now m common use (Fig. 205) are con-
structed on this principle. They contain a compound ribbon of
metal (Fig. 206) wound into a spiral, one end of which a is fixed

EXPANSION-COEFFICIENTS. 267

SO as to be immovable, while the other is attached to a contrivance
for multiplying motion which moves the index. With a rise in tem-
perature, the more expansible metal on the outside produces an
increase of curvature, which causes the spiral to wind up closer.
This motion is communicated to the index, which points on the
dial to the corresponding temperature. With a fall of temperature
the action is reversed.

Advantage is taken of this principle also in the construction of
balance wheels of chronometers. The rate of vibration of a chro-
nometer balance wheel depends upon its mass
and the distance of its circumference from the
center. The parts BC and FG (Fig. 207) are
made up of a compound strip, the more expan-
sible metal being on the outside. As the tem-
perature rises the radii AA expand, and the

chronometer would lose time, but the heat

' Fig. 207.

causes the strip BC and FG lo curve inwards.

The masses D and D' are thus brought nearer the center, and this

compensates for the expansion of A.

Experiment 3. — Fit stoppers tightly in the necks of two similar thin
glass flasks (or test-tubes), and through each stopper pass a glass tube
about 60 cm long. The flasks must be as nearly alike as possible. Fill
one flask with alcohol and the other with water, and crowd in the stop-
pers so as to force the liquids in the tubes a little way above the corks.
Set the two flasks into a basin of hot water, and note that, at the instant
the flasks enter the hot water, the liquids sink a little in the tubes, but
quickly begin to rise, until, perhaps, they reach the top of the tubes and
run over.

When the flasks first enter the hot water they expand, and thereby
their capacities are increased ; meantime the heat has not reached the
liquids to cause them to expand, consequently the liquids sink momen-
tarily to accommodate themselves to the enlarged vessel. Soon the heat
reaches the liquids, and they begin to expand, as shown by their rise in
the tubes. The alcohol rises faster than the water. Different substances,
in both the solid and the liquid states, expand unequally on experiencing
equal changes of temperature.

Experiment 4. — Take a dry flask like that used in Exp. 3, insert
the end of the tube in a bottle of colored water (Fig. 208), and apply
heat to the flask ; the enclosed air expands and comes out through the
liquid in bubbles. After a few minutes withdraw the heat, keeping the

268

MOLECULAR DYNAMICS.

end of the tube in the liquid ; as the air left in the flask cools, its pressure
decreases, and the water is forced by atmospheric pressure up the tube
into the flask, and partially fills it.

Experiment 5. — Partly fill a foot-ball with cold
air, close the orifice, and place it near a fire. The
air will expand and distend the ball.

228. Expansion - coefficients. — The ex-
pansion which attends a rise of tempera-
ture depends not only upon the size of the
body, and upon the number of temperature
degrees through which it is heated, but
upon a quantity peculiar to the substance
itself called its expansion-coefficient. This
term is applied to the increase of unit-length
per degree rise of temperature.

Suppose that a rod of length I at 0° C. be
heated through t degrees, so that its length

becomes l^ \ then, representing the linear expansion-coefficient

by c, we have

Fig. 208.

whence Z^ ^ Z (1 -|- ct>).

The expression 1 -\- ct^ called the expansion- factor., is evi-
dently the ratio of the final to the original length. Hence
Zi == Z (1 + ct) \ that is, multiplying the length of a solid at
0° C. by the expansion factor gives its length at t degrees
above zero. Conversely, dividing its length at t° by the
expansion factor gives its length at 0°.

Table of Mean Coefficients of Linear Expansion between 0°

AND 100° C.

Glass ...... 0.0000085

Platinum 0.0000085

Steel 0.000012

Wrought iron . . . 0.000012

Cast iron ..... 0.000011

Gold . 0.000015

Copper ...... 0.000017

Brass 0.000019

Silver 0.000019

Tin 0.000022

Lead 0.000029

Zinc ....... 0.000029

FORCE IN EXPANSION AND CONTRACTION. 269

In the expansion of fluids we have to do only with increase
of volume, called volume or cuhical expansion. A volume-
expansion-coefficient is the increase of unit volume per degree
rise of temperature. This is approximately 3 c, or three times
the linear expansion-coefficient, and may be taken as such for
most practical purposes. Likewise, the surface or superficial
expansion-coefficient is approximately 2 c.

Not only do the expansion-coefficients of liquids and solids
vary with the substance, but the coefficient for the same
substance varies with the temperature, being greater at high
than at low temperatures. Hence, in giving the expansion-
coefficient of any substance it is customary to give the mean
coefficient through some definite range of temperature, as
from 0° to 100° C.

229. Force exerted in expansion and contraction. — The force
which may be exerted by bodies in expanding or contracting
may be very great, as shown by the following rough calcula-
tion : If an iron bar, 1 sq. in. in section, be raised from 0° C.
(melting point of ice) to 500° C. (a dull red heat), its length,
if allowed to expand freely, will be increased from 1 to 1.006,
its expansion-coefficient being about .000012. Now, a force
of about 90 tons is required to stretch a bar of iron of
1 sq. in. section this amount, and this is very nearly the force
that would be necessary to prevent expansion caused by the
heat. It would require an equal force to prevent contraction
(caused by what ?) if the bar be cooled at 500°.

Boiler plates are riveted with red-hot rivets, which, on
cooling, draw the plates together so as to form very tight
joints. Tires are fitted on carriage-wheels when hot, and, on
cooling, grip them with very great force.

It is to be observed that while the force exerted in expan-
sion and contraction is great, the distance through which it
acts is very small, and hence the quantity of work performed
is not very great.

2T0 MOLECULAR DYNAMICS.

230. Anomalous expansion and contraction. — Water pre-
sents a partial exception to the general rule that matter
expands on receiving heat and contracts on losing it. If a
quantity of water at 0° C, or 32° F., be heated, it contracts
as its temperature rises, until it reaches 4° C, or about 39° ¥.,
when its volume is least, and therefore it has its maximum
density. If heated beyond this temperature it expands, and
at about 8° C. its volume is the same as at 0°. On cooling,
water reaches its maximum density at 4° C, and expands as
the temperature falls below that point.

Water is said to have a negative expansion-coefficient be-
tween 0° and 4° C, or between 32° and 39.2° F. A few other
substances, such as India rubber and iodide of lead, contract
when heated, and have, therefore, negative coefficients.

Section VII.

KINETIC THEORY OF MATTER. LAWS OF GASEOUS BODIES.
ABSOLUTE TEMPERATURE.

231. Kinetic theory of matter. — In the case of solids the
molecules are thought to move in curved orbits the centers of
which are fixed. In liquids the orbits are curved; but, as
shown in the phenomena of diffusion (p. 138), the molecules
have, besides the oscillating motion, a motion of translation.
The theory that the molecules composing all bodies of matter
are in perpetual relative motion is called the kinetic theory of
matter. This theory claims that in gases the molecules are so
far separated from one another that their motions are not
generally influenced by molecular attractions. Hence, in
accordance with the first law of motion, the molecules of
gases move in straight lines and with uniform velocity until
they collide with one another or strike against the walls of
the containing vessel, when, in consequence of their elas-

PRESSURE OF A GAS. ETC. 271

ticity, they at once rebound and start on a new path. We
may picture to ourselves what is going on in a body of calm
air, for instance, by observing a swarm of bees in which
every individual bee is flying with great velocity, first in one
direction and then in another, while the swarm either remains
at rest or sails slowly through the air.

232. P7'essure of a gas due to the kinetic energy of its
molecules. — Consider, then, what a molecular storm must be
raging about us, and how it must beat against us and against
every exposed surface. According to the kinetic theory, the
pressure of a gas (or its expansive force as it is sometimes
called) is entirely due to the striking of the molecules against
the surfaces on which the gas is said to press, the impulses
following one another in such rapid succession that the effect
produced cannot be distinguished from constant pressure.^
Upon the kinetic energy of these blows, and upon the number
of blows per second, must depend the amount of pressure.
But we have learned that on the energy of the individual
molecules depends that condition of a gas called its tempera-
ture; so, it is apparent, as stated above, that the pressure of a
given quantity of gas varies ivith its temperature. Again, as
at the same temperature the number of blows per second
must depend upon the number of molecules in the unit of
space, it is apparent that the j^ressure vai^ies with the density.

' ' If the rarefaction of air can be carried so far that only one
particle out of every million is left in the space exhausted, the mean
path of the particles would then be about 4 inches. In our atmos-
phere at a hight of 210 miles, the particles are relatively so few

1 The following estimates made (by Maxwell, using a proposition formulated by
Clausius) for hydrogen molecules at 0°C., and under a pressure of 760 mm, may
prove interesting :

Mean velocity, 6100 feet per second.

Mean path without collision, 38 ten-millionths of an inch.

Collisions, 17,750 millions per second.

Mass, 216,000 million million million in 1 gram,

Nuniber, 19 million million million fill 1 cubic centimeter.

272

MOLECULAR DYNAMICS.

that each particle might travel through a uniform atmosphere of
that density for sixty million miles without entering into collision."
— Daniell.

233. Expansion and expansive force of gases.

The effect of a change of temperature upon a gas may be meas-
ured by noting the change in its volume when the pressure upon
it is constant, or the change in its pressure when its volume is
unchanged. Conversely, the changes in volume or pressure of a

Fig. 209.

gas may be made to indicate changes in temperature. On this
principle the so-called air-thermometer is constructed.

The relation between pressure and temperature of air kept at a
constant volume may be found by means of an apparatus like that
represented in Eig. 209. A bulb &, whose capacity at 0° and
100° C. is known, is filled with dry air. The capillary tube leading
from the bulb is connected to a tube T, which is connected with

ABSOLUTE ZERO. 273

another tube T' open to the atmosphere. The lower ends of T and
T^ dip into a reservoir of mercury E. The bulb b is first sur-
rounded by melting ice, and by means of the screw S the mercury
is forced to the hight h in the tube T, and the difference in level
between {h and h') the surfaces of mercury in the two tubes is
ascertained. By adding this to the hight of the barometer at the
same moment, the total pressure to which the air in the bulb is
subjected at the temperature (0°C.) of melting ice is ascertained.
The bulb is next introduced into an apparatus for boiling water, as
shown in the figure, and surrounded with steam from the boiling
water. By means of the screw S the mercury is again forced to
the same hight h as before in the tube T. But since the pressure of
air increases with the temperature, the mercury will now be much
higher in tube T' and higher in proportion to the increased pressure.
By this means Regnault ascertained that the pressure of dry air
confined to the same volume is about 1.367 times greater at 100°
than at 0° C. ; in other words, the increase of pressure for 100° is
.367, or (.367 -MOO = ) .00367 per degree.

By a slight modification of this instrument and a variation in the
method of using it, Regnault ascertained that when air is allowed
to dilate with increase of temperature while the pressure remains
constant, the volume at 100° is increased by .367 its volume at 0°.
Dividing this number by 100, we obtain .00367 for the expansion
coefficient of air between 0° and 100° C.

It is found that the expansion-coefficient of all gases is approxi-
mately the same as long as they remain true gases, but as they
approach the vaporous state the coefficient changes rapidly.

234. Absolute zei'o. — The zeros on the thermometrio scales
which we have hitherto considered are provisional, arbitrary.
Absolute zero is the temperature corresponding to total
absence of heat. At the absolute zero the molecules must be
supposed to be at rest. At this temperature gases (if they
may be called such) exert no pressure, and occupy no space
save that which their molecules take up when closely packed
together. The point of absolute zero is independent of the
conventions of man. It is a point of absolute cold or absence
of heat, beyond which no cooling is conceivable.

Now it has been found (§ 233) that the pressure in air

274 MOLECULAR DYNAMICS.

increases or diminislies by .00367 = (about) gi^ of its pressure
at 0° for each centigrade degree of rise or fall of tempera-
ture, the volume being maintained constant. If air were a
perfect gas, and could be cooled down in this way to — 273°
C. ( — 459.4° F.), it would exert no pressure. The reason it
would exert no pressure is that its particles possess no kinetic
energy, no motion. This is assumed, therefore, to be the
absolute zero of temperature.

235. Thermo-dynamic definition of temperature.

In this system, temperature, i, is defined by the equation E = kt,
in which E is the average kinetic energy per molecule of a perfect
gas which has that temperature, and fc is a constant. This is called
the thermo-dynamic definition of temperature.

236. Absolute temperature. — Absolute temperature is that
reckoned from the absolute zero, or — 273° C. Temperatures
Tneasured from absolute zero are proportional to the pressure of
a theoretically perfect gas of constant volume or density ; this
statement is merely a convenient expression of the laws of a
perfect gas (§ 237).

The absolute temperature (based on the above theory) of
any body is found by adding 273 to its temperature as
indicated by a centigrade thermometer, or 459.4 to its tem-
perature as indicated by a Fahrenheit thermometer. The
comparative scale given on p. 257 will make this clear.

237. Laws of gaseous masses. — It follows, from the above
discussion, (1) that the volume of a given mass of gas at con-
stant pressure is proportional to its absolute temperature; i.e. at

V (volume of a sriven mass of eras)

constant pressure — ^ — T^ry-y — . , , r^ — - remains

T (absolute temperature)

constant. This is called the Law of Charles.

(2) The pressure of a given 7nass of gas luhose volume is kept
constant is j^roportional to its absolute temperature.

Boyle's law states that (3) at a constant temperature the
volume of a given mass of gas is iiiversely proportional to its

LAWS OF GASEOUS MASSES. 275

pressitre; i.e. the iiroduct of its jpressure and its volume is con-
stant. NoWj when both the pressure and the volume vary at
the same time, it may be shown that (4) the product of the
pressure and the volume of a given mass of gas is proportional
to its absolute temperature. A gas is said to be perfect, when
it perfectly obeys these laws.

We may also state this law as follows : the product of the
pressure and volume of a given mass of gas divided by its
absolute temperature is a constant quantity, or

in which P = pressure, V^ volume, T= absolute temperature
of a given mass of gas, and 6' = a constant quantity, the value
of which depends on the gas in question.

Exercises.

1. Find, in both centigrade and Fahrenheit degrees, the absolute tem-
peratures at which mercury boils and freezes.

2. At 0° C. the volume of a certain mass of gas under a constant
pressure is 500 cc ; a. What will be its volume if its temperature be
raised to 75° C? 6. What will be its volume if its temperature become
— 20°C.?

3. If the volume of a mass of gas at 20° C. be 200 cc, what will be its
volume at 30° C? Solution: 20° C. is equivalent to (20 + 273) 293 abs.
temp. ; then 293 : 303 : : 200 : 206.8 cc. Ans.

4. To what volume will a liter of gas contract if cooled from 30° C. to
-15°C.?

5. One liter of gas under a pressure of one atmosphere will have what
volume, if the pressure be reduced to 900 g per square centimeter, while
the temperature remains constant ?

6. The volume of a certain mass of air at a temperature of 17° C,
under a pressure of 800 g per square centimeter, is 500 cc ; what will be
its volume at a temperature of 27° C, under a pressure of 1200 g per
square centimeter? Solution: 17° C. is equivalent to 290° abs. temp. ;
27° C. is equivalent to 300° abs. temp. Then 290 : 300 : : 500 X 800 : x X
1200. Whence x = 344.8 cc. Ayis.

7. If the volume of a mass of gas under a pressure of 1 K per square

276 MOLECULAR DYNAMICS.

centimeter at a temperature of 0° C. be 1 liter, at what temperature will
its volume be reduced to 1 cc under a pressure of 200 K per square centi-
meter ? Ans. : 54.6° abs. temp., or — 218.4° C.

8. Find the temperatures on the absolute scale at which the substances
named on p. 278 melt.

9. If a cubic foot of coal-gas at 32° F. , when the barometer is at 30 in. ,
has a mass of 2V ^^-i what will be the mass of an equal volume at 68° F.,
when the barometer is at 29 in. ?

10. Explain the following statement : "To compare absolute tem-
peratures, we may seal up a mercurial barometer in a tube, or an aneroid
barometer in a preserving jar." Whiting.

Section VIII.

EFFECTS OF HEAT. FUSION.

238. Change of pi^operties in solids attending change of tem-
perature. — " Every known property of a piece of matter,
except its mass, varies with variation of temperature." Inas-
much as heat tends to weaken cohesion, the rigidity and
tenacity of solids are generally lessened, and their plasticity
is increased, by the addition of heat.

239. Fusioji. — As previously stated, whether a given sub-
stance exist in a solid, liquid, or gaseous state depends upon
the temperature and the pressure it is under. Solids exposed
to heat liquefy or fuse, unless previously decomposed. Some,
like ice and tin, change their state abruptly ; others, like
glass and wrought iron, become plastic prior to liquefaction.
The temperature at which a substance melts is called its
fusion-point. The fusion-points of different substances vary
greatly : that of alcohol ( — 130.5° C.) and that of iridium
(1950° C.) may be taken as extreme examples.

Experiment and experience teach that (1) the melting or
solidifying point (they are approximately the same for the same
substance) m.ay vary ividely for different substances, but for the
same substance it is invariable when under the same pressure.

CHANGE OF VOLUME DURING SOLIDIFICATION. 277

(2) The temioeratuve of a solid or liquid remains constant at
the nielting-])oint from the moment that melting or solidifteation
begins until it ceases.

Exjperiment 1. — Put a lump of ice as large as your two fists into
"boiling water ; when it is reduced to about J its original size, skim it out.
Wipe the lump, and place one hand on it and the other on a lump to
which heat has not been applied ; you will not perceive any difference in
their temperatures. Under ordinary pressure ice cannot be made warmer
than 0° C.

240. Change of volume during fusion or solidification.
During these changes of state there is usually a change of volume.

Generally solids expand on melting, so that the volume is increased
and the density is diminished by fusion. There are, however,
certain important exceptions to this rule. Ice and type-metal, for
example, float in the denser liquids of these substances respectively.
Such metals as expand on solidifying are especially adapted for
casting, since they expand and fill every portion of the mould, and
the resulting impression is sharp and clear.

241. Effect of pressure on the fusion point.

In solids whose volume increases during fusion, the fusion is more
difficult when the pressure is increased, since not only is heat
required to change the state of the body, but an additional quantity
is required to do additional external work against the additional
pressure. In solids whose volume decreases during fusion, an
increase of pressure facilitates the fusion, and the temperature is
less as the pressure is greater. Briefly, then, pressure lowers the
fusion point of substances that expand on solidifying, and raises the
fusion point of those that contract.

24:2. Regelation.

Experiment 2. — Place in contact the smooth surfaces (wiped
dry) of two pieces of ice ; press them together for a few seconds ;
remove the pressure, and they will be found to be firmly frozen
together.

This phenomenon is called regelation, and is simply the con-
sequence of lowering the fusion point by pressure. The fusion
point being lowered, the ice at the surfaces of contact is melted, but
as the water produced is below the normal fusion point of water, it
freezes again as soon as the pressure is removed.

278

MOLECULAE DYNAMICS.

Table of Edsion Points.

Alcohol -130.5° C.

Mercury . .
Sulphuric acid
Ice . . . .
Phosphorus .
Sulphur . .
Tin ...
Lead . . .

. -38.8°

. -34.4°

0°

44°

115°

about 233°

" 334°

Zinc .

Silver . .
Gold . . .
Cast iron .
Wrought iron
Platinum .
Iridium (the most
infusible metal)

about

425° C.

. . 954°

. 1200°

1050-1250°

1500-1600°

. 1775°

. 1950°

243. Heat of fusion. — The temperature of ice remains
constant while melting, and generally heat imparted to a
melting body affects its temperature very little if any.
Furthermore, ice and other solids are not instantly converted
into liquids on reaching the fusion point, but absorb a quan-
tity of heat proportionate to their mass before fusion is
accomplished. Inasmuch as none of the heat applied during
melting raises the temperature of the body, the question
arises ivhat becomes of the heat applied to the body? The
thermo-dynamical theory furnishes the only satisfactory
answer to this important question. The answer is, about all
the heat applied to a body during fusion is consumed in doing
internal ivot'k, as it is called. The molecules that were held
firmly in their places by molecular forces are, during fusion,
moved from their places, and so work is done against these
forces. Heat, the energy of motion, performs this work, and
is thereby converted into pMential energy, the energy of
position, — energy of the same kind as that of a raised
weight. The heat which disappears in melting is called the
heat of fusion.

If it require a large quantity of heat and a long time to
effect the fusion of a body, it must be inferred that the
amount of work done is proportionately great. Fortunate is
it that it does require much heat to melt moderately small

MEASUREMENT OF THE HEAT OF FUSION. 279

masses of ice and snow, else on a single warm day in winter
all the ice and snow would melt, creating most destructive
freshets.

244. Measurejneyit of the heat of fusion. — Let it be required
to find approximately the quantity of heat that disappears
during the melting of one kilogram of ice. This quantity is
most readily determined by the method of mixtures.

Experiment 3. — Weigh out 200 g of dry ice chips (dry them with a
towel), whose temperature in a room of ordinary temperature may be
safely assumed to be 0° C. Weigh out 200 g of boiling water, whose
temperature we assume to be 100° C. Pour the hot water upon the ice,
and stir it imtil the ice is all melted. Test the temperature of the resulting
liquid.

Suppose its temperature is found to be 10° C. It is evident that the
temperature of the hot water in falling from 100° to 90° would yield
sufficient heat to raise an equal mass of water from 0° to 10° C. Hence
it is clear that the heat which the water at 90° yields in falling from 90°
to 10° — a fall of 80° — in some manner disappears. At this rate had you
used 1 k of ice and 1 k of hot water, the amount of heat lost would be 80
calories. Careful experiments, in which suitable allowances i are made
for loss or gain of heat by radiation, conduction, absorption by the calo-
rimeter, etc. , have determined that 80 calories of heat are consumed in
melting 1 kilogram of ice.

Table ^ of Heats of Fusion of Substances under the Pressure
OF ONE Atmosphere.

Calories Calories Calories

per kilogram. per kilogram. per kilogram.

Water .... 80.0 Silver . . 24.7 Sulphur . . 9.4

Sodium nitrate . 63.0 Tin . . . 14.25 Lead . . . 5.4

Potassium nitrate 47.4 Bismuth . . 12.6 Phosphorus . 5.0

Zinc. .... 28.1 Iodine . . 11.7 Mercury. . 2.82

According to this table it is to be understood, for example,

1 As this is not a manual of manipulation, the cumbersome details relating to
cautions against errors which must be observed in order to secure even approximately
accurate results are omitted. The student will find ample directions in almost all of
the many laboratory manuals extant.

2 Barker,

280 MOLECULAR DYNAMICS.

that one kilogram of ice at 0° "ander tlie pressure of one
atmosphere, while changing to a liquid, absorbs without any
change of temperature 80 calories of heat, or as much heat as
is required to raise 80 kilograms of water from 0° to 1° C.
It will be seen that water possesses the greatest latent heat
of fusion.

245. Transformation of heat reversible. — As stated at the
beginning of this chapter, work is transformable into heat,
and, as stated on p. 87, potential energy is transformed into
kinetic energy '^ by the return of the molecules to their orig-
inal positions ; " so when water freezes or any liquid is
re-solidified, the potential energy (latent heat of fusion) reap-
pears as heat.

Water in freezing undergoes no change of temperature,
hence, if heat be developed during the operation, it must
become diffused or must be " given off " in order to allow the
freezing to go on. As the diffusion is necessarily slow, so
freezing must be slow ; and this slow development of heat
and its immediate dispersion accounts for the fact that we
are seldom made conscious of the development of heat during
solidification.

Farmers sometimes turn to practical use this well-known
phenomenon. Anticipating a cold night, they carry tubs of
water into cellars to be frozen. The heat generated thereby,
although of a low temperature, is sufiicient to protect vege-
tables which freeze at a lower temperature than water.

Heat disappears in the process of melting ice ; and, para-
doxical as it may seem, heat is generated by freezing water.
By freezing one kilogram of water 80 calories of heat are
generated.

VAPORIZATION. 281

Section IX.

EFFECTS OF HEAT CONTINUED. VAPORIZATION.

246. Evaporation ; ebullition. — The process of converting
a liquid into a vapor is called vaporization. A comparatively
slow vaporization which takes place only at the exposed sur-
face of a liquid is called evap)oration. A rapid process which
may take place throughout the liquid, but usually is most
rapid at the point where heat is applied, is called boiling or
ebullition.

247. Kinetic theory of evaporation and condensation. —
According to this theory, some of the molecules in any liquid
move faster than others. Those at the surface which have
great velocities, if the direction of their motion be from the
liquid, will break away from the forces that are able to retain
the molecules moving more slowly, and will fly about as
vapor in the space outside the liquid. This is evaporation.
At the same time molecules of the vapor striking the liquid
may plunge into it and become entangled in it, and thus
there is a return to the liquid state. This is condensation.
The number of molecules which passes from the liquid to the
vapor increases with increase of temperature of the liquid.
The number which passes from the vapor to the liquid de-
pends upon the density of the vapor as well as its tempera-
ture. When the density of the vapor increases to such an
extent that as many molecules are condensed as are evaporated,
then the vapor is said to have its maximum density for that
temperature, or to be saturated. The evaporation then ap-
pears to cease, because the proportions of liquid and vapor
remain unchanged. Liquids which evaporate readily are
called volatile liquids in distinction from those which do not,
and which are called fixed liquids.

248. Boiling point. — In evaporation, molecules fly from

282

MOLECULAR DYNAMICS.

the surface of the liquid and mingle with the particles of the
air and drive only a certain small portion of them away. In
boiling, the molecules which fly away from the surface drive
all the air particles away a certain distance. Hence the
vapor of a boiling liquid must exert a pressure at least
as great as the atmospheric pressure. The greater the
external pressure to be overcome, the greater must be the
energy, i.e. the higher the temperature, of the vapor. When
the saturated vapor of a liquid exerts a pressure equal to that
of the atmosphere, the liquid begins to boil, and the tempera-
ture at which this occurs is called the normal boiling point of
that liquid.

Experiment 1. — Half fill a glass flask with water. Boil the water
over a Bunsen burner ; the steam will drive the air from the flask. With-
draw the burner, quickly cork the flask very tightly, and plunge the flask

into cold water, or invert the flask and
pour cold water upon the part contain-
ing steam, as in Fig. 210 ; the water in
the flask, though cooled several degrees
below the usual boiling point, boils again
violently. The application of cold wa-
ter to the flask condenses some of the
steam, and diminishes the pressure of
the rest, so that the pressure upon the
water is diminished, and the water boils
at a reduced temperature.

If hot water be poured upon
the flask, the water ceases to boil.
Under the receiver of an air-pump,
water may be made to boil at
any temperature between 0° and
100° C. ; indeed, if exhaustion be
carried far enough, boiling and freezing may be going on at the
same time. When high temperature is objectionable, appara-
tus is contrived for boiling and evaporating in a vacuum ; as,

Fig. 210.

BOILING POINT. 283

for instance, in the vacuum pans used in sugar refineries. As
water boils more easily under diminished pressure, so it boils
with more difficulty when the pressure is increased ; and the
temperature to which water may be raised under the pressure
of its own steam is limited only by the strength of the vessel
containing it. Vessels adapted to resisting steam pressure,
called digesters, are often employed to effect a complete pene-
tration of water into solid and hard substances. By this
means gelatine is extracted from the interior of bones. In
the boiler of a locomotive, where the pressure is sometimes
150 lbs. above >the atmos]3here, the boiling point rises to about
185° C. (365° F.).

Experiment 2. — Place a test tube (Fig. 211) half filled with ether in a
beaker containing water at a temperature of 60° C. Although the tem-
perature of the water is 40° below its boiling point, it very
quickly raises the temperature of the ether sufficiently to
cause it to boil violently. Introduce a chemical thermom-
eter into the test tube, and ascertain the boiling point of
ether.

Experiment 3. — Li a beaker half full of distilled water
suspend a thermometer so that the bulb will be covered by
the water and yet be at least two inches above the bottom -p^^ 211
of the beaker. Apply heat to the beaker, and observe any
changes of temperature which may occur, both before and after boiling
begins.

Experiment 4- — Dissolve table-salt in water, and you may raise its
boiling point till it reaches 108° C. With saltpetre it may reach 115° C.

It is found that (1) /or a given pressure (for example, that
of the atmosphere at 760 mm) every liquid has a definite boil-
ing point ; (2) this hoiliiig point remains constant after boiling
has begun; (3) salts dissolved in liquids raise their boiling
points, but do not ajfect the temperature of the escaping vapor.
The latter result is obviously due to the additional work
required to be performed by the heat in order to overcome
the increased cohesion due to the salt in solution.

284

MOLECULAR DYNAMICS.

Boiling Points under

Sulphurous anhydride . — 8° C.

Ether 34.89°

Carbon disulphide . . 46.8°

Bromine 63°

Wood-spirit .... 65.50°

Alcohol 78.39°

A Pressure of 760 mm
Benzole . .
Water . . .
Acetic acid .
Butyric acid .
Sulphuric acid
Mercury . . .

80.44° C.
100°

117.28°
157°
337.77°
358°

Boiling Points of Water at Different Pressures
Barometer
680 mm
700

720
740
760
770

96.9° C.

97.7°

98.5°

99.26°

100°

100.3°

Atmosplieres

100° c.
120.6°
134°

152.2°

180.3^

The boiling point of water varies with the altitude of

places, in consequence of the change in atmospheric pressure.

Eoughly speaking, a difference of altitude of 533 ft. causes a

variation of 1° F. in the boiling point. The measurement of

hights by means of the boiling point is called hy^jsoinetry.

A hypsometer is simply a convenient portable apparatus for

boiling water, provided witli a thermometer sensitive to (say)

0.01°.

Boiling Points of Water at Different Altitudes.

Above the
sea-level.

Quito + 9,500 ft.

Mont Blanc . . . 15,650 "

Mt. Washington . . 6,290"

Boston "

Dead Sea (below) . - 1,316 "

249. Vaporization of solids. — The boiling point of a sub-
stance under ordinary atmospheric pressure may be below its
fusion point; if so, the solid changes directly into a vapor
without passing througli the usual intermediate liquid state.
For example, the maximum vapor pressure of carbon dioxide

Mean higlit of
Barometer.

21.53 in. .

Temperature
. . 91° C.

16.90 " .

. . 86°

22.90 " .

. . 94°

30.00 " .

. . 100°

31.50 " .

. . 101°

HEAT OF VAPORIZATION.

285

at its fusion point (—66° C.) is three atmosplieres ; under
any less pressure it vaporizes, but never melts. Ice cannot be
melted under less than 4.6 mm pressure. It requires a greater
pressure than that of a single atmosphere to melt arsenic and
arsenious oxide. All of the substances named above evapor-
ate far below their fusion points. Hence, arsenious oxide ex-
posed in a room, in wall paper, for example, may render the
air of the room dangerous for inhalation. Ice evaporates at
temperatures far below the fusion point. Housekeepers well
know that clothes dry without thawing, and it is a familiar
fact that snow-banks diminish in size when the weather is
below zero.

250. Heat of vaporization. — Heat that is consumed in the
process of vaporization is called the heat of vaporization.
The quantity of heat required to convert a gram of water at
100° C. into steam without altering its temperature (which is
the same as the quantity of heat generated by the condensa-
tion of one gram of steam at 100°) is called the heat of
vaporization of water.

Experiment 5. — Let it be required to find the heat of vaporization of
water. Find tlie mass in grams of the glass beaker or calorimeter C (Fig. 212),
and since it will receive a small
portion of the heat generated
by the condensation of the
steam, find its water equiva-
lent by multiplying its mass
by the specific heat of glass
(.177). Represent this quan-
tity by mi. Take in the
calorimeter a certain known
mass M of cold water at
a known temperature t.
When water in the flask A
begins to boil, introduce the •^^^' ^-^^^

end of the delivery tube B into the water in C. The steam that passes
through the tube is condensed on entering the cold water, and heats the

286 MOLECULAR DYNAMICS.

water. When a considerable portion of the water in A has been
vaporized, the temperature ti of the water in C is taken again, and the
contents of the calorimeter are again weighed. The increase m in the
mass of water in C is the mass of steam which has been condensed.
Let L be the heat of vaporization. Then the whole quantity of heat
generated by the condensation of m grams of steam is Lm and the quan-
tity of heat imparted to the cold water in falling from 100° to ti° is
m (100 — ti), or the total quantity of heat given to the calorimeter and its
original contents is Lm + m (100 — ^i). The heat required to raise the
calorimeter and its original contents from t to ^i is (ilf +mi) (^i ~t).
But these two quantities are equal, hence

Lm + m (100 - ^i) = (Jf + mi) {h - t);
whence L = (J^ + »»,) (e, - Q - ^(100 - ^

In practice various precautionSj which need not here be
detailed, are necessary. Careful experiments have determined
the value of L for steam to be 536 (Kohlrausch) small
calories ; that is, it requires 536 small calories of heat to
convert one gram of water at 100° into steam at 100°, or
536 calories per kilogram, and when the process is reversed
536 calories per kilogram of steam are generated by the
condensation.

When water is converted into steam, the larger portion of
the heat which disappears is consumed in separating the
molecules so far that molecular attraction is no longer sen-
sible ; a small portion — about Jg — is consumed in over-
coming atmospheric pressure.^ The amount of work done in
boiling is very great, as shown by the amount of heat
consumed. Hence it requires a long time for the water to

1 In this connection a brief discussion of some molecular hypotheses may prove of
interest. The molecule is regarded as a collection of atoms. It may possess both
translatory and rotary motion as a whole, and hence kinetic energy. It may possess
potential energy, in virtue of the mutual attraction between it and other molecules
and their relative distances, since the potential energy increases with both the force
and the distance apart. This latter may be termed Mi^er-molecular energy. The
atoms within the individual molecule may also possess all these kinds of motion and
energy, and the total amount of this energy ivithin the molecule may be called intra-
molecular energy. The work which is performed when either the inter- or intra-

DISTILLATION.

287

acquire the requisite amount of heat. This is a protection
against sudden changes.

Steam is a most convenient vehicle for the conveyance of
heat of vaporization, i.e. jjotential energy, from the boiler to
distant rooms requiring to be heated. For example, for every
kilogram of steam condensed in the pipes of the radiator,
536 calories, or heat enough to raise 5.36 kilograms (about
12 lbs.) of ice-water to the boiling point, are generated.

251. Distillation.

Experiment 6. — Vessel A (Fig. 213), called a condenser^ contains a
coil B, called a worm^ of copper tubing, terminating at one extremity at a.
The other end of the
tube, 6, projects through d \

the side of the vessel
near its bottom. Near
the top of the vessel pro-
jects another tube, c,
called the overflow^ with
which is connected a
rubber tube, e. This
tube conveys the warm
water which rises from
the surface of the heated
worm away to a sink or
other convenient recep-
tacle.

Take a glass flask of
a quart capacity, fill it
three - fourths full of
pond or bog water. Connect the flask by means of a glass delivery-tube

Fig. 213.

molecular energy is altered, or both are altered, is called disgregation tvork. The
heat energy is regarded as the kinetic energy of the molecules in their vibratory
motions, apart from any energy of rotation about their own centers of mass, and
apart from inter- or intra-molecular energy.

When heat is imparted to a body, it is distributed, in general, so as to produce, in
varying relative quantities, four effects, viz. : (1) to raise the temperature by
increasing the vibratory speed of the molecular motion ; (2) to perform external
work ; (3) to perform inter-molecular work ; (4) to perform intra-molecular work.
3 and 4 are classed as disgregation work.

288 MOLECULAR DYNAMICS.

with the extremity a of the worm. Heat the water in the flask ; as soon
as it begins to boil, commence siphoning cold water through a small tube, d^
from an elevated vessel, E, into the condenser. Inasmuch as the worm is
constantly surrounded with cold water, the steam on passing through it
becomes condensed into a liquid, and the liquid (called the distillate)
trickles from the extremity h into a receiving vessel. The distillate is
clear, but the water in the flask acquires a yellowish brown tinge as the
boiling progresses, due to the concentration of impurities (largely of
vegetable matter) which are held in suspension and solution in ordinary
pond water. The apparatus used is called a stilly and the operation
distillation.

When a volatile liquid is to be separated from water, — for
example, when alcohol is separated from the vinous mash after
fermentation, — the mixed liquid is heated to its boiling-
point, which is lower than that of water. Much more of the
volatile liquid will be converted into vapor than of the water,
because its boiling point is lower. Thus a partial separation
is effected. By repeated distillations of the distillate, a 95
per cent alcohol is obtained.

Sbction X.

METHODS OF PRODUCING COLD ARTIFICIALLY.

A body becomes cold only by losing heat. As heat passes
only from warmer to colder bodies, it is evident that the
temperature of a body 'cannot fall below that of surrounding
bodies, — for example, below the temperature of other bodies
in the same room, — by the natural process of imparting heat
to its neighbors. The temperature of a body, then, can be
reduced below that of its neighbors only by some artificial
means.

The fact that heat must be consumed in the conversion
of solids into liquids and liquids into vapors, because work is
done, is turned to practical use in many ways for the

HEAT CONSUMED IN EVAPORATION. 289

purpose of producing artificial cold. The following experi-
ments will illustrate this process.

252. Heat consumed in dissolving. — Freezing mixtures.

Experiment 1. — Prepare a mixture of 2 parts, by mass, of pulverized
ammonium nitrate and 1 part of ammonium chloride. Take about 75 cc
of water (not warmer than 8°C.), and into it pour a large quantity of the
mixture, stirring it while dissolving with a test-tube containing a little
cold water. The water in the test-tube will be quickly frozen. A finger
placed in the solution will feel a painful sensation of cold, and a thermom-
eter will indicate a temperature of about — 10° C.

One of the most common freezing mixtures consists of
3 parts of snow or broken ice and 1 part of common salt.
The affinity of salt for water tends to produce liquefaction of
the ice, and the resulting liquid dissolves the salt, hoth opera-
tions consuming heat.

253. Heat consumed in evaporation. — The heat consumed
in vaporization is greater than that consumed in liquefaction ;
for example, in the case of water it is greater in the ratio of
536 : 80. Hence evaporation is the more efficient means
of producing extremely low temperatures. Whatever tends
to hasten evaporation tends to accelerate the reduction of
temperature. Rapidity of evaporation increases with the tem-
perature., extent of surf ace exposed, diminution of pressure, and
dryness of the atmosphere (see page 292). The more volatile
the liquid employed for evaporation, other things being equal,
the more rapid the consumption of heat.

Experiment 2. — Fill the palm of the hand with ether ; the ether
quickly evaporates, and produces a sensation of cold.

Experiment 3. — Place water at about 40° C. in a thin porous cup,
such as is used in a Grove's battery, and the same amount of water at
the same temperature in a glass beaker of as nearly as possible the same
size as the porous cup. Introduce into each a thermometer. The com-
paratively large amount of surface exposed by means of the porous vessel

290 MOLECULAK DYNAMICS.

will so hasten the evaporation in this vessel, that, in the course of 10 to 15
minutes, a very noticeable difference of temperature will be indicated by
the thermometers in the two vessels.

In warm climates water is frequently kept in porous
earthen vessels in order that its temperature may be kept low
enough by evaporation to render it suitable for drinking.

Experiment 4- — Fill an atomizer, such as is used in the toilet for
throwing a spray of cologne, with ether and throw a spray of the liquid
continuously upon the bulb of a thermometer. In a very short time the
temperature of the mercury will fall, in a warm room, to — 8° C. or lower.

Water may be frozen by its own evaporation in the receiver of an
air-pump from which the air (and consequently the air-pressure) is
removed. A dish of sulphuric acid should be placed in the receiver
to absorb the water-vapor.

By evaporating liquid ethylene and liquid air under a pressure of
4 mm, Olzewski produced a temperature of — 220°C.i The cold
produced by the evaporation of liquid carbon dioxide in the air,
when it is relieved from pressure, is sufficient to freeze the greater
part of it, producing a solid mass like snow, which evaporates
slowly, producing a temperature of — 90° C.

254. Sjjheroidal state,

A drop of water placed on a smooth metal surface heated above
200° C. will not come in contact with or moisten the surface, but
assumes the form of a fiat spheroid and rolls about like a ball or
spins on its axis. It is said to be in the spheroidal state. When in
this state the liquid does not boil ; indeed, its temperature is several
degrees below its boiling point. The liquid globule rests upon a
cushion of its own vapor and is buoyed up by it. When^ however,
the heated metal cools and the vapor pressure is not great enough
to sustain the globule, it comes in contact with the metal, its tem-
perature rapidly rises to the boiling point, and it is quickly con-
verted into steam. Showmen place red hot irons in their mouths
and dip their moistened hands into melted lead or even melted iron,

1 See Philosophical Magazine (Feb. 1895). Tlie announcement is just made in
Nature that Olzewski has lately succeeded in liquefying hydrogen and producing a
temperature of — 243° C.

DEW-POINT. 291

without injury. A layer of spheroidal fluid prevents contact of the
flesh with the heated metal.

Boutigny placed liquid sulphur dioxide, whose temperature when
in the spheroidal state is below zero, in a red-hot platinum crucible ;
it quickly assumed the spheroidal state, and drops of water let fall
upon it quickly froze. Mercury can in like manner be frozen in a
red-hot crucible by employing liquid nitrous oxide in the spheroidal
state.

Section XI.

HYGROMETRY.

255. Dew-point. — Hygrometry treats of tlie state of the air
with regard to the water vapor it contains. A given space,
e.g. a cubic meter (it matters little whether there is air in the
space or whether it is a vacuum), — can hold only a limited
quantity of water vapor. This quantity depends on the tem-
perature. The capacity of a space for water vapor increases
rapidly with the temperature, being nearly doubled by a rise
of 10° C. On the other hand if air containing a given quan-
tity of water vapor be cooled, it will continually approach and
finally reach saturation, since the lower the temperature, the
less the capacity for water vapor. It is evident that air
saturated with vapor cannot have its temperature lov^ered
without some of the vapor being condensed into a liquid,
which will appear, according to location and condition of
objects within it, as dew, fog, or cloud} The temperature at
which this condensation occurs is called the dew-point for air
containing this proportion of water vapor. The dew point
may be defined as the temperature of saturation for the
quantity of water vapor actually present in the air. The

-i Clouds formed at temperatures above 0° consist of minute spherical drops of
water, ^^Lq to -^^-^ of an inch or more in diameter. Clouds formed at temperatures
below 0° consist of minute ice spicules, which may increase in size and become
snow-flakes.

292

MOLECULAR DYNAMICS.

greater the quantity of water vapor present in tlie air,
the higher is its dew-point. Capacity for water vapor depends
upon temperature ; dew-point depends upon quantity of vapor
present.

If the existing temperature be far above dew-point it in-
dicates tliat the air can contain much more vapor than there
is in it at the time, and the air is said to be dry. If the tem-
perature of the air be little above
dew-point, the air is said to be
humid, which means that it
can hold but little more vapor.
The sensation of dryness experi-
enced, especially in rooms heated
artificially, does not depend
upon the absolute quantity of
water vapor present per cubic

r"^'"" """T|"" "^L_|y foot.
J The heat of a stove, for in-

^ stance, dries the air of a room

without destroying any of its
water vajDor. In such a room,
the lips, tongue, throat, and skin
experience a disagreeable sensa-
tion of dryness, owing to the
rapid evaporation which takes
place from their surfaces. This
should be taken as nature's ad-
monition to keep water in the stove urns, and in tanks con-
nected with furnaces.

The quantity of water vapor present in the air is expressed
either (1) by the mass of vapor per unit of volume ; or (2) by
the ratio between the quantity actually present and that which
would be present if the air were saturated at the temperature
of observation. The latter is the more common and more

Fig. 214.

WET AND DRY BULB THERMOMETER.

293

useful method, aud this ratio is called the relative humidity,
or simply " humidity '^ of the air. It is expressed in per-
centages. Thus, relative humidity = 75 per cent., or 0.75,
denotes that the air contains three-fourths the quantity of
water vapor required to saturate it at the present tempera-
ture.^

256. Wet and dry hulh thermometer.

The relative humidity of air is measured in various ways and by
various devices. The instrument most commonly used is Mason's
wet and dry bulb thermometer, or as it is frequently called, psy-
chrometer. It consists (Fig. 214) of two thermometers mounted side
by side a short distance apart, one having a dry bulb and the other
a bulb covered with muslin, kept moist by capillary action through

1 The student may be profited by a perusal of the following, copied from the daily
meteorological report in tbe London Times. [Speed the time when our own govern-
ment, in consideration of its educational value at least, shall issue daily a similarly
complete meteorological bulletin.]

The Times Office, 2 a.m.

Keadings of Jordan Barometer (cor-
rected) DURIXG THE PAST TWENTT-
FOUR HOURS.

Temperature and Htgrometric Condition of the
Air in Londox. August 25, 26.-

i August 25-26. 1891. •
1 A.M. P.M. ^

To^hef^ ^

t 6

S10>

:'i

I'. 2

iUi

321-
320-
319-
31&
317^

:29-8
729-7
;29-6
:29-5

Hours

Obser-
vation.

Tempe
Air.

rature.

Dew
Point.

Tension

of
Vapour.

Weight of

Vapour

inlO

cubic feet

of air.

Drying
Power of
Air (per
10 cubic
feet).

Humidity
(Satura-
tion =
100)-

Noon..
9p.m..
2 a.m..

Degrees.
62
63
62

Degrees.
52
54

Inches.
•388
■■118
■433

Grains.
13
16

Grains.
19

18
14

Per Cent.
69

72
77

Minimum Temperature, 55 deg. Maximum Tempera-
ture, 63 deg.

BEN NEVIS OBSERVATORY, Aug. 25.
Summit Station, (4,40/ ft. above sea level).

Bar.

Temperature.

Wind.

Cloud.

At 320

Dry
Bulb.

Wet
Bulb.

Direc-
tion.

Force.
to 6.

Species.

Amount
to 10.

9 A.M.
9 P.M.

In.
24^820
24-605

Deg.
37-4
42-5

Deg.
Sat.
Sat.

S.W.
S.W.

Mist
Mist

10
10

Maxinmm temperature, 43 5; minimum temperature, J
Black bulb, 56. Sunshine, none. Rainfall, 0-905 in.
Base Station (42 ft. above sea level).

Bar.

Temperature.

Wind.

Cloud.

At 32°

Dry
Bulb

Wet
Bulb

Direc-
tion.

Force
0to6

Species.

Amount
tolO

9A.M
9 P.M.

In.
29-189
28-891

Deg.
53-8
53-9

Deg.
52-8
53-6

S.
Calm

Cumulus
Nimbus

10
10

Maodmum temperature, 57-4; minimum temperature, 51-3.
Black bulb, 110. Sunshine, 1 hour 19 min. Rainfall,
0-520 in.

294 MOLECULAR DYNAMICS.

conducting threads of lamp-wick from a vessel of water below. The
dry bulb indicates the temperature of the air itself ; while the wet
bulb, cooled by evaporation, shows usually a lower temperature
according to the amount and rapidity of evaporation. The differ-
ence in temperature of the two bulbs is greatest when the air is
dry est.

By experiment has been ascertained the relative humidity
corresponding to 1, 2, etc. degrees difference between the two
thermometers for any given temperature of the air. Empirical
psychrometrical tables similar to the reduced table in the Appendix
(p. 622) accompany this instrument. The observer reads the tem-
perature of the air and ascertains the difference of the temperatures
of the two bulbs, and from these two numbers determines by the
table the relative humidity, and the dew-point of the air at that
time and place.

Not only does hygrometry play an important part in the science
of meteorology, and consequently have important bearings upon
many branches of industry, but it also has an intimate relation to
the hygienic qualities of the atmosphere. The human body is much
affected by the hygrometric state of the air.

Section XII.

DIFFUSION OR TRANSFERENCE OF HEAT.

There is always a tendency to equalization of temperature ;
that is, heat has a tendency to pass from a warmer body to a
colder, or from a warmer to a colder part of the same body,
until there is an equality of temperature.

There are commonly recognized three processes of diffusion
of heat, — conduction, convection, and radiation.

257. Conduction.

Experiment 1. — Place one end of a wire about 10 inches long in a
lamp-flame, and hold the other end in the hand. Heat gradually travels
from the end in the flame toward the hand. Apply your flngers succes-
sively at different points nearer and nearer the flame ; you find that the
nearer you approach the flame, the hotter the wire is.

CONDUCTION. 295

The flow of heat through an unequally heated body, from
places of higher to places of lower temperature, is called
conduction; the body through which it travels is called a
conductor. The molecules of the wire iii the flame have their
motion quickened; they strike their neighbors and quicken
their motion ; the latter in turn quicken the motion of
the next ; and so on, until some of the motion is finally
communicated to the hand, and creates in it the sensation of
heat.

Experiment 2. — Fig. 215 represents a board on which are fastened,
by means of staples, four wires : (1) iron, (2) copper, (3) brass, and (4)
German silver. Place a lamp-flame where the wires
meet. In about a minute run your fingers along
the wires from the remote ends toward the flame,
and see how near you can approach the flame on
each without suffering from the heat. Make a list
of these metals, arranging them in the order of their
conductivity.

Experiment 3. — Go into a cold room, and place ^iq 215

the bulb of a thermometer in contact with various
substances in the room ; you will probably find that they have the same,
or very nearly the same, temperature. Place your hand on the same
substances ; they appear to have_ very different temperatures. This is
due to the fact that some substances conduct heat away from the hand
faster than others. Those substances that appear coldest are the best
conductors. If you go into a room warmer than your body, all this is
reversed ; those substances which feel warmest are the best conductors,
because they conduct their own heat to your hand fastest.

You learn that some substances conduct heat much more
rapidly than others. The former are called good conductors,
the latter poor conductors. Metals are the best conductors,
though they differ widely among themselves.

Experiment 4. — Nearly fill a test-tube with water, and hold it some-
what inclined (Fig. 216), so that a flame may heat the part of the tube
near the surface of the water. Do not allow the flame to touch the part

296 MOLECULAR DYNAMICS.

of the tube that does not contain water. The water may be made to
boil near its surface for several minutes before any change of the tem-
perature at the bottom will be perceived.

Liquids, as a class, are poorer con-
ductors than solids. Gases are much
poorer conductors than liquids. It is
difficult to discover that pure, dry air
possesses any conducting power. The
poor conducting poAver of our clothing
is due partly to the poor conducting
power of the fibers of the cloth, but
Fig. 216. chiefly to the air which is confined by it.

Loose garments, and garments of loosely woven cloth, inas-
much as they hold a large amount of confined air, furnish a
good protection from heat and cold. Bodies are surrounded
with bad conductors, to retain heat when their temperature is
above that of surrounding objects, and to exclude it when
their temperature is below that of surrounding objects. In
the same manner double windows and doors protect from
cold.

258. Convection in. gases. — Conduction takes place gradu-
ally and slowly at best from particle to particle, the body
and its particles being relatively at rest. Convection takes
place when the body moves or there is relative motion be-
tween its parts, thus carrying heat.

Experiment 5. — Hold your hand a little way from a flame, beneath,
on the side of, and above the flame. At which place is the heat most
intense ?

Experiment 6. — Cover a candle-flame with a glass chimney (Fig. 217),
blocking the latter up a little way so that there may be a circulation of
air beneath. Hold smoking touch-paper near the bottom of the chimney ;
the smoke seems to be drawn with great rapidity into the chimney at the
bottom ; in other words, the office of the chimney is to create what is
called a draft of air. Notice whether the combustion takes place any
more rapidfy with than without the chimney.

CONVECTION IN GASES.

297

Experiment 7. — Place a candle within a circle of holes cut in the cover
of a vessel, and cover it with a chimney, A (Fig. 218). Over an orifice
in the cover place another chimney, B. Hold a roll of smoking touch-
paper over B. The smoke descends this chimney, and passes through
the vessel and out at A. This illustrates the method often adopted to
produce a ventilating draft through mines. Let the interior of the tin
vessel represent a mine deep in the earth, and the chimneys two shafts
sunk to opposite extremities of the mine. A fire kept burning at the
bottom of one shaft will cause a current of air to sweep down the other
shaft, and through the mine, and thus keep up a circulation of pure air
through the mine.

Fig. 217.

Fig. 218.

The cause of the ascending currents is evident. Air, on becoming
heated, expands rapidly and becomes much rarer than the surrounding
colder air ; hence it rises much like a cork in water, while cold air pours
in laterally to take its place. In this manner winds are created. Sea
and land breezes are convection currents.

The so-called trade-ivinds originate in the torrid or heated
zone of the earth. The air over the heated surface of the
earth rises, and the colder air from the polar regions flows in

298 MOLECULAR DYNAMICS.

on botli sideS; giving rise to a constant wind from the KE.^
in the northern hemisphere, and a wind from the S.E. in the
southern hemisphere. Convection currents on the surface of
the sun often attain a velocity of 100 miles per second.

259. Change of temperature in vertical currents ascending
from the earth.

The lower air in contact with the heated surface of the earth
acquires a certain temperature and a corresponding expansive force
previous to its ascent. As it reaches higher altitudes, the pressure
upon it becomes less ; it therefore expands, pushing away the sur-
rounding air, until, as a result of its expansion, its expansive force
is reduced to equality with the pressure upon it. It follows from
the dynamical theory of heat, that in doing this work the ascending
air must expend some of its energy ; i. e. the work is done by the
expenditure of some of its heat ; hence, the ascending air is cooled
by the very processes involved in its ascent. The rate of cooling
thus produced is about 1° for 100 m of ascent. Such changes are
called adiahatic; i.e. they are produced without a transfer of
heat. *

260. Ventilation. — Intimately connected with the topic
convection, is the subject (of vital importance) ventilation,
inasmuch as our chief means of securing the latter is through
the agency of the former. The chief constituents of our
atmosphere are nitrogen and oxygen, with varying quantities
of water vapor, argon, carbon dioxide gas, ammonia gas, nitric
acid vapor, and other gases. The atmosphere also contains in
a state of suspension varying quantities of small particles of
free carbon in the form of smoke, microscopic organisms, and
dust of innumerable substances. All of these constituents
except the first four are called imjmrities. Carbon dioxide
is the impurity that is usually the most abundant and most
easily detected ; so it has come to be taken as the measure
of the purity of the atmosphere, though not itself the most

1 The easterly component is due to tlie earth's rotation.

VENTILATION.

299

deleterious constituent. Its chief harm arises from its diluent
effect upon the life-giving oxygen. Pure out-door air contains
about 4 parts of carbon dioxide by volume in 10,000. If the
quantity rise to 10 parts, the air becomes unwholesome.

Experiment S. — Ascertain by means of Wolpert's air-tester i (ap-
proximately) the number of parts in 10,000 of carbonic acid gas (and
thereby determine approximately the degree of pollution by
respiration and combustion) in the air of the school-room.

Clean the test-tube with water containing a little vinegar,
and afterwards rinse thoroughly with clean water. Fill the
clean test-tube with lime-water, even with the horizontal
mark. Expel all the air possible from the rubber bulb A
(Fig. 219), by pressing on it with the thumb ; then allow it
to fill with air from the room. Insert the small glass tube
B into the lime-water nearly to. the bottom. Expel the
air in the bulb again with moderate rapidity, so that the
bubbles may rise nearly to the top of the tube C ; but do
not allow the liquid to overflow. Continue the pressure
until you have withdrawn the tube from the liquid, when
you will allow the bulb to refill with air of the room. At
the end of each expulsion place the bottom of the test-
tube .on a sheet of white paper in good daylight, and look
vertically down through the liquid at the black mark on the
bottom of the test-tube ; repeat the process, being careful
not to affect the result more than is necessary with your
breath, until the turbidity of the lime-water renders the
mark invisible. " If the mark become obscured after filling „ ^^
the bulb ten or fifteen times only, the air of the apartment
is unfit for continuous respiration. With good air the bulb must be filled
twenty-five times and upwards. The normal amount in pure out-door
air is 3 to 5 parts per 10,000."

Ascertain, by comparing your results with those given by Prof. Wol-
pert in the table below, the number of parts of carbon dioxide in 10,000
of the air of your room. Kepeat the experiment, taking air at different
elevations in the room.

1 This instrument is vised almost exchisively by inspectors of scliool and other
public buildings in the State of Massachusetts.

300

MOLECUIiAR DYNAMICS.

Number

Carbon

Number

Carbon

Number

Carbon

Number

Carbon

dioxide

Fillings.

per 10,000.

Fillings.

per 10,000.

Fillings.

per 10,000.

Fillings.

per 10,000.

200.

12.5

6.4

4.3

100.

12.

6.3

4.2

67.

11.

6.1

4.1

50.

10.5

5.9

4.1

40.

10.

5.7

4.0

33.

9.5

5.5

3.9

29.

9.1

5.4

3.9

25.

8.7

5.3

3.8

22.

8.3

5.1

3.7

20.

3.7

18.

7.7

4.9

3.6

16.

7.4

4.8

3.5

15.

7.1

4.6

3.5

14.

6.9

4.5

3.4

13.

6.6

4.4

3.3

Carbon dioxide is about one and one-half times heavier than
air at the same temperature ; consequently, when both have
the same temperature, and the former is very abundant, it
tends to sink beneath the air, in which large quantities of
this gas are generated.

The knowledge of this fact has led many to suppose that a
means for the escape of impure air need be provided only
near the floor of a room. But it should be remembered that
(1) the tendency of carbon dioxide, unless present in excessive
quantities, is to diffuse itself equally through a body of air ;
but (2) when it is heated to a temperature above that of the
surrounding air, as when generated by flames, or when it
escapes in the warm breath of animals, it is lighter than the
air, and consequently rises. If this impure air could escape
at the ceiling while fresh air entered at the floor, the ventila-
tion would be good. But usually this fresh air must be
warmed; and in passing over a stove, furnace, or steam
radiator, its temperature will generally become higher than

VENTILATION.

301

that of the impure air, so that it Avill rise above the latter,
and pass out at a ventilator in the ceiling, leaving the floor
cold ; hence, in high school-rooms the most impure air is
often found half-way up.

The quantity of fresh air introduced must be great enough
to dilute the impurities till they are harmless. An adult
makes about 18 respirations per minute, expelling from his
lungs at each expiration about 500 cc of air, over 4 per cent
of which is carbon dioxide. At this rate, about 9,000 cc of air
per minute become unfit for respiration ; and to dilute this
sufiiciently, good authorities say that about 100 times as much
fresh air is needed ; or, for
proper ventilation, about a
cnhic meter of fresh air per
minute is needed for each per-
son^ or, in British measures,
2,000 cubic feet per hour.

Roof

'/,/,/,W,/,lM

Room

Pure outdoor air.

Fig. 220 represents a scheme
for heating a room by steam,
and ventilating it by con-
vection. Steam is conveyed
by a pipe from the boiler
to a radiator box just be-
neath the floor of the room.
The air in the box becomes
heated by contact with and
radiation from the coil of
pipe in the box, and rises
through a passage opening
into the room by means of a
register near the floor at C, a
supply of pure air being kept
up by means of a tubular
passage opening into the box
from the outside of the building. Thus the room is furnished with
"pure warm air, which, mmgiing with the impurities arising from the
respiration of its occupants, serves to dilute them and render them

Fig. 520.

302 MOLECULAR DYNAMICS.

less injurious. At the same time, the warm and partially vitiated
air of the room passes through the open ventilator A into the
ventilating-flue, and escapes, so that in a moderate length of time
a nearly complete change of air is effected. It is evident that on
the coldest days of winter the convection is most rapid ; indeed, it
may be so rapid that the air cannot be heated sufficiently to render
the room near the floor comfortable. At such times the ventilator
A may be closed, while the ventilator B is always open. The
heated air rises to the top of the room and, not being able to escape,
crowds the colder air beneath out at the ventilator B. No system
of ventilation dependent wholly on convection is adequate properly
to ventilate crowded halls ; air is too viscous and sluggish in its
movements. In such cases ventilation should be assisted by some
mechanical means, such as a blower or fan, worked by steam or
water power.

261. Convection in liquids.

Experiment 9. — Fill a small (6 ounce) thin glass flask with boiling hot
water, color it with a teaspoonful of ink, stopper the flask, and lower it
deep into a tub, pail, or other large vessel filled with cold water. With-
draw the stopper, and the hot, rarer, colored water will rise from the
flask, and the cold water will descend into the flask. The two currents
passing into and out of the neck of the flask are easily distinguished.
The colored liquid marks distinctly the path of the heated convection
currents through the clear liquid and makes clear the method by wdiich
heat, when applied at the bottom of a body of liquid, becomes rapidly
diffused through the entire mass notwithstanding that liquids are poor
conductors.

Experiment 10. — Ffll again the flask with hot colored water, stopper,
invert, and introduce the mouth of the flask just beneath the surface of a
fresh pail of cold water. Withdraw the stopper with as little agitation
of the water as possible. What happens ? Explain.

Ocean currents, e.g. the gulf stream, are convection currents.
Liquids are also cooled by convection currents. When the
air above the surface of a pond, for instance, is cooler than
the surface v^^ater, the latter gives heat to the former, cools,
becomes denser, and sinks. Meanwhile the warmer and rarer
water below rises, and in this way the eutire body is kept at

RADIATION. 303

an approximately uniform temperature until it reaches 4° C,
at which point convection ceases.

262. Radiation. — In radiatio7i a hotter body loses heat,
and a colder body is warmed, through the transmission of
undulatory motion in a medium called the ether, ivhich is not
itself heated thereby. It is neither a mass nor a molecular
transference of heat; in fact heat itself is not transferred
by radiation at all. Heat generates radiation (ether wave
motions) at one place, and the body which obstructs these
waves transforms the energy of their motion, or as it is com-
monly called radiant energy, into heat. In this manner the
earth is heated by the sun, though no heat passes between
them. In this manner radiant energy passes through glass
and slabs of ice without heating them much, since they offer
little obstruction to the passage of ether waves. All bodies
emit radiant energy, and there is an exchange of energy
between bodies by radiation, going on at all times. This
mode of transmission of energy is the most important of all,
and will be treated fully in the next chapter.

Section XIII.

THERMO-DYNAMICS.

263. Thermo-dynamics defined. — TherTno-dynamics treats
of the relation between heat and mechanical ivork. One of the
most important discoveries in science is that of the equivalence
of heat and ivork; that is, that a definite quantity of mechanical
work, ivhen transformed ivithout luaste, yields a definite quantity
of heat ; and conversely, this heat, if there be no luaste, can 'per-
form the original quantity of mechanical work.

264. Transformation, correlation, and conservation, of energy.
— The proof of the facts just stated was one of the most
important steps in the establishment of the grand twin con-

304 MOLECULAR DYNAMICS.

ceptions of modern science : (1) that all kinds of energy are
so related to one another that energy of any hind can he trans-
formed into energy of any other kind, — known as the doctrine
of CORRELATION OF ENERGY ; (2) that when one form of
energy disappears, its exact equivalent in another form, ahuays
takes its place, so that the sum total of energy is unchanged, —
known as the doctrine of conservation of energy.

These two doctrines are admirably summarized by Maxwell
as follows : "^The total energy of any body or system of bodies is
a quantity which can neither be increased nor di7iiinished by
any mutual action of these bodies, though it may be transforvied
into any of the forms of which energy is susceptible.'''' Since all
bodies of matter in the universe constitute a system, it
follows from the above that the sum total of energy in the
universe is a constant quantity. Neither creation nor annihi-
lation of energy is possible through any agency known to
man. These doctrines constitute the corner stones of modern
physical science. Chemistry teaches that there is a conser-
vation of matter, i.e. that matter is neither creatable nor
annihilable through any known natural agency or process.

265. Jolliers experiment. — Two laws of ther mo-dynamics. —
The experiment to ascertain the " mechanical value of heat,"
as performed by Dr. Joule of England, was conducted about
as follows :

A copper vessel, B (Fig. 221), was provided with a paddle
wheel (indicated by the dotted lines), which rotated about a
vertical axle, A. The axle was rotated by the weights E and
F, the cord of each being so arranged that each weight, in
falling, rotated the axle in the same direction. By turning
the crank above A the weights are raised to any desired hight
measured on the scales G and H.

The resistance offered by the water to the motion of the
paddles was the means by which the mechanical energy of
the weights was converted into heat, which raised the tem-

JOULE S EXPERIMENT.

305

perature of the water. Taking two bodies whose combined
mass was, e.g.^ 80 K, he raised them a measured distance, e.g.
53m high; by so doing 4240 kgm of work were performed upon
them, and consequently an equivalent amount of energy was
stored up in them, ready to be converted, first into that of
mechanical motion, then into heat. He took a definite mass
of water to be agitated, e.g. 2 K, at a temperature of 0° C.
After the descent of the weights, the water was found to have
a temperature of 5° C. ; consequently the 2 K of water must
have received 10 calories of heat (careful allowance being

Fig. 221.

made for all losses of heat), which is the number of calories
that is equivalent to 4240 kgm of mechanical energy ; or one
calorie is equivalent to 4^4 %^^ (commonly taken as 4.2 X 10''
ergs per gram degree) of mechanical energy.

In other words, to produce the quantity of heat required to
raise 1 kilogram of water through 1° C, 4^4- hilogr ammeters of
mechanical energy must he consumed. What the experiment
really shows is that whenever a certain quantity of mechanical
energy is converted into heat, the number of thermal units
produced is always proportional to the mechanical energy
consumed, or to the work done. This is embodied in the

306 MOLECULAR DYNAMICS.

first law of thermo-dynamics^ which is expressed as follows ■.
" When equal quantities of ^mechanical effect are produced by
a?ii/ means whatever from 2)ureli/ thermal sources, or lost in
purely thermal effects, equal quantities of heat are put out of
existence, or are generated.'''' It is apparent that heat, being a
form of energy, may be measured in ergs. In this way the
erg is regarded as the mechanical nnit of heat. The advan-
tage of this is found in the fact that we are often in the
position of having to solve problems in which heat and work
enter as terms to be added together. The existence of
quantitative correlations between all the various forms of
energy imposes upon men of science the duty of bringing all
kinds of physical quantities to one common scale of com-
parison, as is attempted in the absolute system.

Mechanical energy can be wholly converted into heat, but
it can be demonstrated that heat under the jnost favorable
circumstances conceivable, even with the use of an ideally
perfect heat engine {i.e. one which wastes no heat), can never
be wholly converted into work. A portion — a large portion
— of the heat employed must be given up to some substance
termed technically a '' refrigerator,'' which in some form is a
necessary adjunct to every heat-engine, and that portion still
exists as heat. This is a practical deduction from the so-
called Second Law of Thermodynamics ; ^ viz., " It is impos-
sible to derive m^echanical effect from any portion of m^atter by
cooling it below the temperature of the coolest surrounding ob-
jects. ^^ The ratio of the heat converted into work and the
entire heat employed is called the efficiency of the engine.
" For any boiler-pressure " (of a steam-engine) " which it is
safe to employ in practice, it is not possible, even with a
perfect engine, to convert into work more than about fifteen
percent, of the heat used." ^ — Anthony and Bkackett.

1 Any adequate discussion of this law would take us beyond the limits proposed
for this book. This subject is more fully treated in the works of Barker and Daniell.

MECHANICAL EQUIVALENT OF HEAT.

307

266, Meclianical equivalent of heat. — As a converse of the
above it may be demonstrated by actual experiment that the
quantity of heat required to raise 1 K of water from 0° to
1° C. will, if converted into work, raise a 424 K weight 1 m
high, or 1 K weight 424 m high. According to the British
system, the same fact is stated as follows : The quantity of
heat that will raise the temperature of 1 pound of water from
60° to 61° F. will raise 772.55 pounds 1 foot high. The
quantity, 424 kgm, is called the mechanical equivalent of one
calorie, or Joule'' s equivalent (abbreviated simply J). J is
the number of units of energy or work per unit of heat. Or we
may say that one calorie is the thermal equivalent of 424 kgm
of work,^ or the thermal equivalent of 1 kgm is ^^^ calorie.

Temperature,

Equivalent in

Temperature,

Equivalent in

Centigrade.

Kilogrammeters.

Centigrade.

Kilogrammeters.

0°

431.1

23°

426.0

5°

429.8

25°

425.8

10°

428.5

27°

425.6

15"

427.4

29°

425.5

17°

427.0

31°

425.6

19°

426.6

33°

425.7

21°

426.2

35°

425.8

If we denote by H the number of calories, and by W the
number of kilogrammeters of mechanical energy, then the

HI W

ratio — =r - (a constant) = ^^^ ; whence H = — .

1 A knowledge of the exact numerical value is of great scientific and practical
importance. The results as obtained by Eowland (1879) with improved apparatus
and by improved methods, though the same in principle as that employed by Joule,
are doubtless more accurate and are likely to come into general use for engineering
and scientific purposes.

" The following table gives the number of kilogrammeters required to raise one
kilogram of pure water from f to t° + 1, as found by Rowland, for the latitude of
Baltimore, and at sea-level. At Baltimore, ^ = 980.05 cm. To reduce to any other
latitude chan Baltimore, add, for lat. 30°, 0.34 kgm ; lat. 40°, 0.08 kgm ; lat. 50°,
— 0.41 kgm. The value of J ordinarily used in engineering computations is 424 kgm.
As most measurements with which this value of J is employed are made at about 15°
to 25° C, this value is too small by one-half per cent, or more."

308 MOLECULAE, DYNAMICS.

Section XIY.

thermodynamics continued. steam-engine.

267. Description of a stPMin-engiyie. — A steam-engine is a
macliine in which the elastic force of steam is the motive
agent. Inasmuch as the elastic force of steam is entirely due
to heat, the steam-engine is properly a heat engine ; that is, it
is a machine by means of which heat is continuously trans-
formed into work, or the energy of mass motion.

The modern steam-engine consists essentially of an arrange-
ment by which steam from a boiler is conducted to each side
of a piston alternately; and then, having done its work in
driving the piston to and fro, is discharged from each side
alternately, either into the air or into a condenser. The
diagram in Fig. 222 will serve to illustrate the general fea-
tures arid the operation of a steam-engine. The details of the
various mechanical contrivances are purposely omitted, so as
to present the engine as nearly as possible in its simplicity.

In the diagram, B represents the boiler, F the furnace, S
the steam-pipe through which steam passes from the boiler to
a small chamber VC, called the valve-chest. In this chamber
is a slide-valve V, which, as it is moved to and fro, opens and
closes alternately the passages M and N leading from the
valve-chest to the cylinder C, and thus admits the steam
alternately each side of the piston P. When one of these
passages is open, the other is always closed. Though the
passage between the valve-chest and the space in the cylinder
on one side of the piston is closed, thereby preventing the
entrance of steam into this space, the passage leading from
the same space is open through the interior of the valve, so
that steam can escape from this space through the exhaust-
pipe E. Thus, in the position of the valve represented in the
diagram, the passage N is open, and steam entering the cylin-

DESCRIPTION OF A STEAM-ENGiNE.

309

der at the top drives the piston in the direction indicated by
the arrow. At the same time the steam on the other side of
the piston escapes through the passage M and the exhaust-
pipe E. While the piston moves to the left, the valve moves
to the right, and eventually closes the passage N leading
from the valve-chest and opens the passage M into the same,
and thus the order of things is reversed.

Fig. 222.

Motion is communicated by the piston through the piston-
rod E, to the crank G, and by this means the shaft A is
rotated. Connected with the shaft by means of the crank H
is a rod W which connects with the valve V, so that, as the
shaft rotates, the valve for the greater part of its stroke is
made to slide to and fro, in a direction opposite to that of
the motion of the piston.

310 MOLECULAR DYNAMICS.

The shaft carries a fly-wheel W. This is a large, heavy
wheel, having the larger portion of its mass located near
its circumference ; it serves as a reservoir of energy, which
is needed to make the rotation of the shaft and all other
machinery connected with it uniform, so that sudden changes
of velocity resulting from sudden changes of the driving
power or resistances may be avoided. By means of a belt
passing over the wheel W motion may be communicated
from the shaft to any machinery desirable.

268. Condensing and non-condensmg-engines} — Sometimes
steam, after it has done its work in the cylinder, is conducted
through the exhaust-pipe to a chamber Q, called a condenser,
where, by means of a spray of cold water introduced through
a pipe T, it is suddenly condensed. This water must be
pumped out of the condenser by a special pump, called tech-
nically the air-iDumjp ; thus a partial vacuum is maintained.
Such an engine is called a condensing-engine. Its advantage
is obvious, for if the exhaust-pipe, instead of opening into a
condenser, communicate with the outside air, as in the non-
condensing engine^ the steam is obliged to move the piston
constantly against a resistance arising from atmospheric
pressure of 15 pounds for every square inch of the surface of
the piston. But in the condensing engine a large portion of
the pressure on the exhaust side of the piston is removed and
an equivalent portion of the pressure on the steam side is
utilized and made to do useful work. Tn well proportioned
condensing apparatus the pressure on the exhaust side may
be reduced 90 per cent., so that the moving piston instead of
working against a resistance of 15 lbs. meets with a resistance
of only 1.5 lbs. per square inch.

269. Steam gauge. ■ — An instrument called a steam gauge
is connected with the boiler. It ^measures the excess of the

1 The terms, low-pressure and high-pressure engines, are not distinctive as applied
to engines of tlie present day.

COMPOUND OR DOUBLE-CYLINDER ENGINE.

311

pressure of the steam at any instant above the atmospheric
pressure. The absolute pressure of the steam (i.e. measured
from zero) is the pressure indicated by the steam gauge ^^^i^s
the pressure of the atmosphere at the time.

270. Compound condensing or douhle-cylinder engine. — This
engine has two cylinders, each like that of a simple engine.
One, A (Fig. 223), called the high-pressure cylinder^ receives
steam of very high pressure directly from the boiler through

Fig. 223.

the orifice V. The steam, after it has done work in this
cylinder, passes through the steam-port E into cylinder B,
called the loiu-pressure cylinder. Cylinder B is larger than
cylinder A. The steam which enters cylinder B possesses
considerable pressure, and is therefore capable of doing con-
siderable work under suitable conditions. It should be borne
in mind that in order that steam may do work in any cylinder,
it is necessary that an inequality in the pressure of the steam

312 MOLECULAR DYNAMICS.

each, side of the piston should be maintained ; just as an
inequality of level, i.e. a head, is essential to water-power.
The steam, after it has done its work in cylinder B, passes
through a port C into a condenser (not represented in the
figure), where it is suddenly condensed or let down to a very
low pressure. If a vertical glass tube were led from the con-
denser to a vessel of mercury below, the mercury would
ordinarily stand about 25 inches high in the tube, which
would show that the pressure of the steam against which the
steam when it enters cylinder B does work, is only about
one-sixth of an atmosphere. Much energy is economized by
the compound engine.

271. The locomotive. — The distinctive feature of the loco-
motive engine is its great steam-generating capacity relatively
to its size and weight, which are necessarily limited. To do
the work ordinarily required of it, from three to six tons of
water must be converted into steam per hour. This is
accomplished in two ways : first, by a rapid combustion of
fuel (from a quarter of a'ton to a ton of coal per hour);
second, by bringing the water in contact with a large extent
(about 800 square feet) of heated surface. The fire in the
" fire-box " A (Plate II) is made to burn briskly by means of a
powerful draft which is created in the following manner :
The exhaust steam, after it has done its work in the cylinders
B, is conducted by the exhaust-pipe C to the smoke-box D,
just beneath the smoke-stack E. The steam, as it escapes
from the blast-pipe F, pushes the air above it, and drags by
friction the air around it, and thus produces a partial vacuum
in the smoke-box. The external pressure of the atmosphere
then forces the air through the furnace grate and hot-air
tubes G-, and thus causes a constant draft. The large extent
of heated surface is secured as follows : The water of the
boiler is brought not only in contact with the heated surface
of the fire-box, but it surrounds the pipes G (a boiler usually

POWER OF A STEAM ENGINE. 313

contains about 150). These pipes are kept hot by the heated
gases and smoke, all of which must pass through them to the
smoke-box and smoke-stack.

272. Fower of a steam engine. — The horse-power of a
steam engine is calculated by means of the following formula,

(Mean effective pressure in lbs. per sq. in. on piston X area
of piston in sq. in. X length of stroke in ft. X number of strokes
per min.) -r- 33,000.

The steam engine, with all its merits and with all the
improvements which modern mechanical art has devised, is
an exceedingly wasteful machine. The best engine that has
been constructed utilizes less than 15 per cent, of the heat
energy generated by the combustion of the fuel.

Questions.

1. What kind of engine {i.e. condensing or non-condensing) is that
which produces loud puffs ? What is the cause of the puffs ?

2. Why does the temperature of steam suddenly fall as it moves the
piston ?

3. What do you understand by a ten horse-power steam-engine ?

4. Upon what does the power of a steam-engine depend ?

5. Is the compound engine a condensing or a non-condensing engine ?
Which is the locomotive engine ?

6. The area of a piston is 500 square inches, and the average unbal-
anced steam pressure is 30 pounds per square inch ; what is the total
effective pressure ? Suppose that the piston travels 30 inches at each
stroke, and makes 100 strokes per minute ; 40 per cent being allowed for
wasted energy, what power does the engine furnish, estimated in horse-
powers ?

7. A leaden bullet of mass 56 g strikes a target with a velocity of 300
meters per second ; its temperature is 16° C. If two-thirds of the energy
of the bullet be used in raising its temperature, determine its final tem-
perature.

8. Can ice at 0° C, and under ordinary atmospheric pressure, have its
temperature raised ? Explain.

314 MOLECULAR DYNAMICS.

9." Eind the resulting temperature (C.) of the following mixtures: —
a. 5 K of snow at 0° with 25 K of water at 28°.
6. 4 K of ice at — 10° with 30 K of water at 50°.
c. 10 K of iron at 200° with 2 K of ice at 0°.

10. How many thermal units are required to change 5K of ice at
— 10° C. into water at 10° ?

11. If 30 g of steam at 100° C. be passed into 400 g of ice-water at
0° C, what will be the temperature of the mixture ?

12. A building is heated by hot-water pipes. How does heat get from
the furnace of the boiler to a person in the building ?

13. A building is heated by steam pipes. How does heat get from the
furnace to objects in the building ?

14. A rod of copper at 0° C. measures 10 ft, ; its length at 100° C. is
0.191 inch greater. Find the coefficient of expansion of copper.

15. A silver rod at 0° C. is 10 ft. long ; find its length at 100° C.

16. A cubic meter of air at 100° C. is cooled down to 0°, and at the
same time its pressure is halved ; determine its new volume.

17. A copper ball weighing 3K, taken out of a furnace and plunged
into 8 k of water at 10° C, heated the water to 25° ; find the temperature
of the furnace.

18. If the heat yielded by 1 K of water in cooling down from 100° to
0° C. were employed in heating 10 K of mercury, initially at 20°, to what
temperature would the mercury be raised ?

19. A kilogram of ice at 0°C. is thrown into 6.3 K of water at 15°;
when the ice is melted, the temperature of the water is 2°. Eind the
heat of fusion of ice.

20. A mass of 93.3 g of copper at 80° C. is immersed in 560 g. of water
at 10°, and raises the temperature of the water to 20° ; find the specific
heat of copper.

PART III.

ETHER DYNAMICS.

CHAPTER I.
ENERGY OF ETHER-STRAIN. RADIANT ENERGY. LIGHT.

Section I.

INTRODUCTION.

Owing to the peculiarity of the subject to be treated in
this, the third and final natural division of Physics, it is
deemed expedient to state at the outset some leading proposi-
tions, whose truth must be assumed as the basis for the study
of a large group of natural phenomena. The demonstrations
of the validity of these several assumptions must, however,
be deferred to their proper place in connection with the study
of the phenomena themselves.

273. The ether. — We know matter by its properties as
perceived by means of our senses ; in other words the exist-
ence of any form of matter is to us only an inference from
the phenomena to which it gives rise. By evidence of pre-
cisely similar nature are we led to believe in the existence of
a medium called the ether, pervading all space and penetrating
between the molecules of matter, which are imbedded in it
and surrounded by it as the earth is surrounded by its atmos-
phere. We cannot see, hear, feel, taste, smell, exhaust, weigh,
or measure it, and yet all this, paradoxical as it may seem,

316 ETHER DYNAMICS.

furnislies absolutely no evidence that it does not exist. Briefly
stated, the proof of its existence is this : it furnishes the
basis for the sole conceivable explanation of very many physical
phenomena.

Phenomena occur just as they would occur if all space were
filled with an intangible and invisible medium capable of
transmitting motion and energy, and we can account for all
these phenomena on no other hypothesis ; hence our belief in
the existence of the medium. The evidence of the existence
of ether is as strong and direct as that of the existence of
air. The eye is an ether sense-organ just as the ear may be
called an air sense-organ, or the hand a sense-organ for the
appreciation of grosser forms of matter.^ The ether is a
Tnedium for the transmission of energy in the form of vibrations.

In its structure the ether is assumed to be excessively fine-
grained, ^'Differing from water, glass and metals in being
very much more finely grained in its structure " ^ (Lord Kelvin).
It possesses rigidity,^ and in this respect is like a solid. Bodies
of matter, even so large as the planets, pass freely through it,
encountering little resistance ; therein it is like a perfect
fluid. It is almost perfectly elastic and incompressible.

274. Radiation. Radiant energy. — The transmission of
energy by means of periodic disturbances in the ether is called
radiation; energy so transmitted is called radiant energy;

1 " Instead of beginning by saying tbat we know nothing about the ether, I say
that we know more about it than we do about air or water, glass or iron, — it is far
simpler ; there is far less to know. Its natural history is far simpler than that of
any other body." — Lord Kelyin, in lectures on Molecular Dynamics at Johns
Hopkins University (1884).

2 " The ether is practically a homogeneous solid, — in other words an exceedingly
fine-grained solid, so finely grained that it is practically homogeneous for portions
exceedingly small in linear dimensions in comparison with the wave-length. But no
degree of smallness will dispense with the to and fro motion of the elastic solid rela-
tively to the imbedded molecules."

3 Calculation leads us to infer that its rigidity is about lO-^ that of steel, and its
density 936 x IQ-^i that of water at 4° C, or equal to that of our atmosphere at a hight
of 210 miles, — a density vastly greater than that of the same atmosphere ' in the
interstellar spaces. — Maxwell.

EFFECTS OF RADIANT ENERGY. 317

and the body emitting energy in tliis manner is called a
radiator. The precise nature of the periodic disturbances,
whether they be due to changes of position in the ether, or to
alternation between opposite conditions (e.g. such as succes-
sive local states of strain or distortion and release therefrom)
is unknown to us. We do know, however, that the laws
according to which these changes take place are those of
wave-motion. Space is traversed at all times and in all direc-
tions by myriads of ether-waves of all possible lengths. The
all-pervading ether can be set in vibration by the motion of the
molecules of ordinary matter. This local disturbance creates
ether-waves, and by these waves energy is transferred from
place to place by the process, as stated above, called radia-
tion. Eadiant energy can be transformed into any other
form of energy, and therefore offers no exception to the doc-
trine of correlation of energy.

Just how vibrations of particles of matter create ether-
waves, and what constitutes a wave of ether, are things of
which our knowledge is as yet very deficient. It must be
remembered that ether is a substance very unlike ordinary
matter, and, therefore, reasoning by analogy must often fail.
It is generally supposed that ether-waves are not waves of
compression and rarefaction, like those of sound-waves in air.

Furthermore, the vibrations which occur in the ether are
not longitudinal like those of the air particles during the
passage of sound-waves, but are transversal and somewhat
analogous to the motions of particles of water in water-waves.
That is, the vibrations in ether are at right angles to the
direction in which the wave is propagated, and are therefore
parallel to the wave-front.

275. Effects of radiant energy. — When radiant energy is
received upon the surfaces of our bodies, warmth is felt ; when
upoUjthe bulb of a thermometer, rise of temperature is indi-
ca;ted ; when by the eye, the sense of sight may be affected ;

318 ETHER DYNAMICS.

if, upon sensitive photographic plates, upon the leaves of
plants, and upon various chemical mixtures, chemical changes
may be promoted. Thus it seems that when ether-waves
impinge upon objects .their energy is transformed, producing
effects of different kinds, which are determined by the nature
of the body upon which they fall. The effect which most
concerns us is that produced when the radiations strike the
eye and become the means, through this organ, of awakening
in the brain the sensation which we call Light.

Section II.

LIGHT.

276. Light defined. Hypotheses. — Physiologically speak-
ing, light is the sensation of sight. Physically considered, it
is that agent which, by its action on the retina of the eye, excites
in us the sensation of vision. Two leading hypotheses ^ regard-
ing the nature of light have been propounded, which are
totally different in character. One is the so-called eynission or
corpuscular hypothesis which was supported by Descartes
(1629), Newton (1672), and most physicists up to the early
part of the present century. It assumes that a luminous
body {e.g. the sun) emits minute material particles (cor-
puscles) which travel through space in all directions with
immense velocity ; these particles by their impact upon the
nerve-woven retina produce the sensation of sight. As a
rose emits minute particles which, reaching the nostrils,
enable us to smell the rose, so a star is supposed to emit par-

■ 1 The Platonists maintained that the sensation of light was produced and vision
effected by something which was emitted from the eye to tlie object, and the sense of
vision was explained by the analogy of touch. " The light from the sun, the twink-
ling of the stars, the colors of the rainbow, and the various hues of the floor of
nature remain the same as when they gladdened the heart of Noah ; but how have
the explanations of the phenomena varied ! "

LUMINOUS AND ILLUMINATED OBJECTS. 319

tides of light which, on reaching the eye, enable us to see
the star.

This hypothesis is now discarded by scientists ; the reasons
for its abandonment will appear further on. The theory
which obtains at the present time, called the undulatory or
wave-theory,^ is based upon the hypothesis that energy is
transmitted from body to body, e.g. from the sun to the earth
(and the reverse), in the form of vibrations or wave-action in
the all-pervading ether. In this connection it should be
borne in mind that the evidence of the correctness of any
theory is its exclusive competence to explain and coordinate
phenomena. It is not claimed that all phenomena have been
fully explained by the wave theory; it is merely claimed
that all we know at the present time about light is in perfect
accord with it. It will be observed that both theories recog-
nize the fact that light is essentially dynamic. According to
the latter theory, light is that vibration of the ether which may
be appreciated by the organ of sight. '^

277. Luminous and illuminated objects. — Some bodies are
seen by means of light-waves which they generate and emit ;
e.g. the sun, a candle flame, and a '' live coal " ; they are
called luminous bodies. Other bodies are seen only by means
of light-waves which they receive from luminous ones and
reflect to the eye, and, when thus rendered visible, are said to
be illumhiated ; e.g. the moon, a man, a cloud, and a "dead''
coal.

1 The first person who presented the wave-theory of light in a definite shape was
Huygens, in a work published in 1690 under the title of Traite de la Lumiere. The
theory was thoroughly established by Young and Fresnel between the years 1800 and
1820.

2 It will be shown further on, that not all ether-waves are capable of afEecting
the sight, hence for the purpose of distinction we apply the term light-waves to those
ether-waves only which are capable of producing vision. It is strongly recommended
that the student in beginning this branch of science make use of the term Hght-icaves
instead of light except when such usage would lead to an inconvenient circumlocu-
tion, in order that he may have strongly impressed upon his mind the fact that when
he is dealing with light he is dealing with waves.

320

ETHER DYNAMICS.

278. Light itself invisible. — Light makes visible to us
luminous or illuminated objects, light-waves from which
actually reach our eyes ; but if we look across the line of
direction of a series of light-waves, termed the path of the
light, we cannot see the light. If we appear to see a sun-
beam admitted through a key-hole or knot-hole, and travers-
ing a darkened room, it is only because it is made to reveal its
track by illuminating the dust motes floating in the air. If
the air in a certain space be cleansed of dust, the path of a
sunbeam through the space will be totally imperceptible.^

279. Light-waves travel in straight
lines. — The path of light-waves ad-
mitted into a darkened room through
a small aperture, as indicated by the
illuminated dust, is perfectly straight.
An object is seen by means of light-
waves which it sends to the eye. A small
object placed in a straight line between
the eye and a luminous point may in-
tercept the light-waves in that path,
and the point become invisible. Hence
we cannot see around a corner, or
through a bent tube.

280. Ray, beam, pencil. — Any line
EE (Fig. 224) which pierces the surface
of an ether-wave ab perpendicularly, is
called a ray. The term '' ray " is but

an expression for the direction in which motion is propagated,
and along which the successive effects of ether-waves occur. '^ If
the wave-surface a^V be a plane, the rays R'E' are parallel,

1 See Tyndall's Fragments of Science, p. 277.

2 In dealing with certain phenomena {e.g. reflection of light) we may, to facilitate
our study, consider the light as propagated, in straight lines or rays ; but we must
hear in mind that a ray has no material or physical existence, for it is a wave that
is propagated, not a ray.

TRANSPARENT AND OPAQUE SUBSTANCES. 321

and a collection of such rays is called a beam. If the wave-
surface <:t"6" be spherical, the rays E."R" have a common point
at the center of curvature ; and a collection of such rays is
called a 'pencil.

281. Transparent, translucent, and opaque substances. — Sub-
stances are transparent, translucent, or opaque, according to
the manner in which they act upon the light-waves which
are incident upon them. Generally speaking, those sub-
stances are transparent that allow other objects to be seen
through them distinctly, e.g. air, glass, and water. Those
substances are translucent that allow light-waves to pass, but
in such a scattered condition that objects are not seen dis-
tinctly through them, e.g. fog, ground glass, and oiled paper.
Those substances are opaque that apparently cut off all the
light-waves and prevent objects from being seen through
them. When bodies intercept light, they are said to cast
shadows.

282. Every point of a luminous body an independent source
of light-waves. — Place a candle flame in the center of a
darkened room ; each wall and every point of each wall
becomes illuminated. Place your-
self in any part of the room, i.e.
in any direction from the flame ;
you are able to see not only the
flame, but every point of the
flame ; hence every point of the
flame must emit light-waves in
every direction. Every point of
a luminous body is an independent
source of light-iuaves, and emits
them in every direction. Such a
point is called a luminous point.

In Figure 225 there are represented a few of the infinite
number of pencils of light emitted by three luminous points

322

ETHER DYNAMICS.

of a candle flame. Every point of an illuminated object ab
receives light from every luminous point.
283. Images formed through small apertures.

Exjperiment 1. — Cut a hole about 8^™ square in one side of a box ;
cover the hole with tin-foil, and prick a hole in the foil with a pin.
Place the box in a darkened room, and a candle flame in the box near
the pin hole. Hold an oiled-paper screen before the hole in the foil ; an
inverted image of the candle flame will appear upon the translucent
paper. An image is a kind of picture of an object.

If light-waves from objects illuminated by the sun — e.g.
trees, houses, clouds, or even an entire landscape — be allowed
to pass through a small aperture in a window shutter and strike
a white screen (or a white wall) in a dark room, inverted images
of the objects in their true colors will appear upon the screen.
The cause of these phenomena is easily understood. When no

screen intervenes between the
candle and the screen A (Fig.
226), every point of the screen
receives light from every point
of the candle ; consequently,
at every point on A, images of
the infinite number of points
of the candle are formed. The
result of the confusion of
images is that no image is distinguishable. But let the screen
B, containing a small hole, be interposed ; then, since light
travels only in straight lines, the point Y' can receive an
image only of the point Y, the point Z' only of the point Z,
and so for intermediate points ; hence a distinct image of the
object must be formed on the screen A. That an image may
he distinct, the images of different jjoints of the object must not
mix, and therefore all rays from each point on the object must
he carried to the corresponding point on the image.

A
if

4J,

■ — '

Fig. 226.

SHADOWS. 323

The brightness of the image decreases as the opening is
made smaller, since less light can pass through it. The
aperture, if small, may have any shape without affecting
the outline of the image. The image of the sun is a circle,
irrespective of the shape of the aperture, if its rays strike
the screen perpendicularly ; but elliptical, if they strike the
screen obliquely.

284. Shadows.

Experiment 2. — Procure two pieces of tin or cardboard, one 18^™
square, the other S^m square. Place the first between a white wall and
a candle flame in a darkened room. The opaque tin intercepts the light
that strikes it, and thereby excludes light from a space behind it.

This space is called a shadow. That portion of the surface
of the wall that is darkened is a section of the shadow, and
represents in form a cross section of the body that intercepts
the light. A section of a shadow is frequently for conven-
ience called a shadow. Notice that the shadow is made up
of two distinct parts, — a dark center bordered on all sides
by a much lighter fringe. The dark center is called the
umbra., and the lighter envelope is called the penumbra.

Experiment 3. — Carry the tin nearer the wall, and notice that the
penumbra gradually disappears and the outline of the umbra becomes
more distinct. Employ two candle flames, a little distance apart, and
notice that two shadows are produced. Move the tin toward the wall,
and the two shadows approach each other, then touch, and finally over-
lap. Notice that where they overlap the shadow is deepest. This part
gets no light from either flame, and is the umbra ; while the remaining
portion gets light from one or the other, and is the penumbra.

Just so the umbra of every shadow is the part that gets no
light from the luminous body, while the penumbra is the part
that gets light from some portion of the body, but not from the
whole.

324

ETHER DYNAMICS.

Experiment 4. — Repeat the above experiments, employing the smaller
piece of tin, and note all differences in phenomena that occur. Hold a
hair in the sunlight, about a centimeter in front of a fly-leaf of this book,

and observe the shadow cast by the
P — ^ ""^ hair. Then gradually increase the dis-

tance between the hair and the leaf,
and note the change of phenomena.

If the source of light were a single
luminous point, as A (Fig. 227), the
shadow of an opaque body B would be of infinite length, and would con-
sist only of an umbra. But, if the source of light have a sensible size,
the opaque body will intercept just as many separate pencils of light as
there are luminous points, and consequently will cast an equal number of
independent shadows.

Let A B (Fig. 228) represent a luminous body, and C D an opaque
body. The pencil from the luminous point A will be intercepted between

Fig. 227.

Fig. 228.

the lines C F and D G, and the pencil from B will be intercepted between
the lines C E and D F. Hence, the light will be wholly excluded only
from the space between the lines C F and D F, which enclose the umbra.
The enveloping penumbra, a section of which is included between the
lines C E and C F, and between D F and D G, receives light from certain
points of the luminous body, but not from all.

LIGHT REQUIRES TIME TO PASS THROUGH SPACE. 325

Questions.

1. Why are images formed through apertures inverted ?

2. Why is the size of the image dependent on the distance of the
screen from the aperture ?

3. Why does an image become dimmer as it becomes larger ?

4. Why do we not imprint an image of our person on every object in
front of which we stand ?

5. Upon what fact does a gunner rely in taking sight ?

6. Explain the umbra and penumbra cast by the opaque body H I,
Fig. 228.

7. When will a transverse section of the umbra of an opaque body be
larger than the object itself ?

8. When has an umbra a limited length ?

9. Wbat is the shape of the umbra cast by the sphere C D, Eig. 228 ?

10. If C D should become the luminous body, and A B a non-luminous
opaque body, what changes would occur in the umbra and the shadow
cast?

11. Wliy is it difficult to determine the exact point on the ground
where the umbra of a church-steeple terminates ?

12. What is the shape of a section of the shadow cast by a circular
disk placed obliquely between a luminous body and a screen ? What is
its shape when the disk is placed edgewise ?

13. The section of the earth's umbra on the moon in an eclipse always
has a circular outline. What does this show respecting the shape of the
earth ?

14. Describe the shadow cast by the earth.

15. Wliy does the electric arc lamp cast well defined shadows ?

Section III.

SPEED OF LIGHT.

285. Light requires time to pass through space. — That light
travels with finite speed was first established in 1676 by the
Danish astronomer Olaf Eoemer, then engaged in Paris in
observing the eclipses of Jupiter's moons. He made obser-

326 ETHER DYNAMICS.

vations on a certain one of Jupiter's satellites wMch. revolves
round this planet as the moon does round the earth. At
regular intervals the satellite ^oasses behind the planet
and is eclipsed within its shadow. The observed intervals,
however, were found to be shorter than the mean value
when the Earth and Jupiter were approaching each other,
and longer when they were receding from each other. It
was evident that this difference was due to the time con-
sumed by the light in crossing the intervening spaces.
From the results of these observations it was calculated
that light required 16 minutes and 36 seconds to traverse the
diameter of the earth's orbit, approximately 185,000,000
miles.

It was then an easy matter for Eoemer to determine how
far light travels per second. The speed of light as deter-
mined by Eoemer is 192,500 miles per second. It has been
determined by later experiments and more reliable methods
that this estimate is too great. The result obtained by
Michelson at Cleveland (1882) is 299,853 kilometers (= about
186,380 miles) per second. This may be accepted as probably
the nearest approximation yet made to the true speed of light
in a vacuum. At this rate, light would encircle our earth
between seven and eight times in a second.

Sound creeps along at the comparatively slow pace of about
one-fifth of a mile (or -J- Km) per second. The former is the
speed with which waves in ether are transmitted ; the latter,
the speed with which waves in air move forward. This great
difference can be accounted for only on the supposition that
the ether is far less dense and much more elastic than air.

Notwithstanding its great speed, light requires no less
than three years to reach us from the nearest fixed star
(a Centauri), and from those more distant it requires cen-
turies. It is thus possible, through the instrumentality of
light, faintly to conceive of the vastness of space.

UNIT OF MEASUREMENT. 327

Section IY.

intensity of illumination.

286. Unit of measurevient. — The unit generally employed
for the measurement of the intensity of the light emitted by
a luminous body is the British candle poiver} It is the in-
tensity of light emitted by a sperm candle -J in. in diameter,
burning 120 grains to the hour.

287. Diminution of intensity of illuminating capacity ivith
distance. A2)2^lication of the law of inverse squares to light.
— Light diminishes in intensity, and hence in its power to
illuminate objects which it strikes, as it recedes from its
source. The intensity of light diminishes as the square of the
distance from its source increases. Calling the quantity of
light falling upon a visiting card at a distance of 2 feet from
a lamp flame 1, the quantity falling upon the same card at a
distance of 4 feet is \, at a distance of 6 feet it is i, and so
on. This is the meaning of the law of inverse squares, as
applied to light.

1 The French unit is the carcel (the name given to a lamp), which is equal to 9J
candles. The unit adopted at the International Congress of Electricians in 1884 is
the light emitted by a square centimeter of molten platinum at the temperature of
solidification, or about 2.08 carcels, or 19.8 candles. This is called the platinum
standard ; and the method, the Yiolle method. Subsequently (1889), o^(j of this unit
was adopted as the practical standard.

" By photometric methods it is found that the sun gives us 1575 billions of billions
times as much light as a standard candle would do at that distance.

" The intensity of sunlight, or the intrinsic brightness of the sun's surface, is
quite a different. matter from the total quantity of its light expressed in candle
power. By intensity we mean the amount of light per square unit of luminous
surface. From the best data we can get we find that the sun's surface is about
190,000 times as bright as that of a candle flame ; and about 150 times as bright as
the lime of a calcium light.

" The brightest part of an electric arc comes nearer sunlight in intensity than
anything else that we know, being from one-half to one-quarter as bright as the solar
surface itself."— Young's Elements of Astronomy.

" If there were an electric light of 2000-candle power on each square foot of the
surface of the earth, the whole light from the earth would be less than one billionth
that from the sun." — Lajstgley. The earth intercepts an extremely small part of
the whole quantity of light emitted by the sun.

328

ETHER DYNAMICS.

Fig. 229.

This law may be illustrated thus : A square card placed
(say) 1 foot from a certain point in a candle flame, as at A
(Fig. 229), receives from this point a
certain quantity of light. The same
light if not intercepted would go on to
B, at a distance of 2 feet, and would
there illuminate four squares, each of
the size of the card, and being spread
over four times the area can illuminate
each square with only one fourth the intensity. If allowed
to proceed to C, 3 feet distant, it illuminates nine such
squares, and has but one ninth its intensity at A. The law
is strictly true only when distance from individual points is
considered.

288. Fhotometry. — The law just established enables us to
compare the illuminating power of one light with that of an-
other, and to express by numbers their relative illuminating
powers. The process is called ijhotometry (light-measuring) ;
and the instrument employed, 2^ fliotometer.

II \/ 1

„.^,.

'^0 '"

60 ' -i

^,U|U,,y,M,,M,,^

(100

^_^

Fig. 230.

289. The Bunsen photometer (Fig. 230) has a screen of
paper, S, mounted in a box, B, open in front and at the
two ends. The box slides on a graduated bar. The screen
has a circular central spot saturated with paraffine, which
renders the spot more translucent than other portions of the
screen. One side of the screen is illuminated by the light,
L, whose intensity is to be measured, and the other side by

THE BUNSEN PHOTOMETER.

329

a standard candle, L'. When the screen is so placed that
the two sides are equally illuminated by the two lights, the
paraffined spot becomes nearly invisible. When one side is
more strongly illuminated than the other, the spot appears
dark on that side and light on the other. The candle power
of the two lights is directly proportional to the square of their
respective distances from

the screen when it is
equally illuminated on
both sides.

In order to render both
sides of the disk simul-
taneously visible, two
mirrors, m and m' (Fig.

Fig. 231.

231), are placed in the box in a vertical position so as to
reflect images of the circular spot in the screen, S, to the
eyes at E Ei.

Questions.

- 1. Suppose that a lighted candle is placed in the center of each of
three cubical rooms, respectively 10, 20, and .30 feet on a side ; would a
single wall of the first room receive more light than a single wall of either
of the other rooms, or less ?

2. Would one square foot of a wall of the third room receive as much
light as would be received by one square foot of a wall of the first room ?
If not, what difference would there be, and why the difference ?

3. If a board 10 cm square be placed 25 cm from a candle flame, the
area of the shadow of the board cast on a screen 75 cm distant from the
candle will be how many times the area of the board ? Then the light
intercepted by the board will illuminate how much of the surface of the
screen if the board be withdrawn ?

4. Give a reason for the law of inverse squares.

5. To what besides light has this law been found applicable ?

6. The two sides of a paper disk are illuminated equally by a candle
flame 50 cm distant on one side and a gas flame 200 cm distant on the
other side. a. Compare the intensities of the two lights at equal dis-
tances from their sources. 6. If the candle be a standard candle, what
is the intensity of the gas flame ?

330 ETHER DYNAMICS*

Section Y.

apparent size of an object.

290. Visual angle.

Experiment. — Prick a pin-hole in a card, place an eye near the hole,
and look at a pin about 20 cm distant. Then bring the pin slowly toward
the eye, and the dimensions of the pin will appear to increase as the dis-
tance diminishes.

Why is this ? We see an object by means of its image
formed on the retina of the eye ; and its apparent magnitude
is determined by the extent of the retina covered by its
image. Rays proceeding from opposite extremities of an
object, as AB (Fig. 232), meet and cross each other within

Fig. 232.

the eye. Now, as the distance between the points of the
blades of a pair of scissors depends upon the angle that the
handles form with each other, so the size of the image
formed on the retina depends upon the size of the angle,
called the visual angle, formed by these rays as they enter
the eye. But the size of the visual angle diminishes approxi-
mately as the distance of the object from the eye increases,
as shown in the diagram ; e.g. at twice the distance the angle
is about one-half as great ; at three times the distance the
angle is one-third as great ;and so on. Hence, distance affects
the aioparent size of an object. Our judgment of the size of
objects is, however, influenced by other things besides the
visual angle which they subtend.

MIRRORS.

331

Section VI.

REFLECTION OF LIGHT.

291. Miry^oj's. Images. — Objects having polished surfaces
which reflect light regularly (i.e. do not scatter the light),
and show images of objects presented to them, are called
mirrors. The mirror itself, if clean and smooth, is scarcely
visible. An image is a picture of an object. According to
their shape mirrors are called planey concave^ convex, spherical,
parabolic, etc.

Experiment 1. — a. Look at the mirror M through the hole marked O
in the metal band (Fig. 233). You see in the mirror an image of the
hole through which you look,
but you do not see the image
of any of the other holes. Eays
that pass through this hole

strike the mirror perpendicu-
larly and are said to be normal
to the mirror. Rays falling
upon an object are called inci-
dent rays. The point where a ray strikes is called the point of incidence.
The reflected rays in this case are thrown back in the same lines and
through the same hole that the incident rays travel. Bays normal to a
mirror after reflection simply retrace their own course, h. Next hold a
candle flame at one of the other holes, e.g. at the hole marked 10. You
can see the image of the candle flame only through the hole of the same
number and at an equal distance on the other side. The angle which an
incident ray makes with a line normal at the point of incidence is called
the angle of incidence, and the angle made by a reflected ray with the
normal is called the angle of reflection.

Law of Reflection. The angles of incidence and reflec-
tion are in the same plane, and are equal.
292. The luave-theory applied to reflection.

The following is an explanation of reflection in accordance with
the wave-theory. Suppose KA, ND, etc. (Fig. 234) to be parallel
rays of a beam of light falling on a plane mirror HI. KLMN

332

ETHER DYNAMICS.

may represent the plane front of one of the waves. As soon as the
wave reaches A, that point becomes the origin of a disturbance in
the ether, which spreads out in the form of a sphere having its
center at A. This disturbance may for convenience be called an
undulation. Let the arc of a circle described around A as a center
denote the boundary which the undulation has reached during the
interval between the arrival of the plane-wave at A and D respec-
tively ; then the radius of this circle is equal to the excess of ND

over KA, because so long as light travels in the same isotropic
medium its speed in all directions is the same. Similarly let circles
be described around points B, C, and J) with radii determined in
the same manner. A straight line a&cD, drawn tangent to these
circles at the points a, 6, etc. , represents a plane reflected wave-front
corresponding to the plane incident- wave. It is inclined to H I at
the same angle but in the opposite direction from a normal.

293. The doubled angle of reflection.

When a mirror is rotated, a beam
of light reflected from it is deflected
through an angle equal to twice that
of the rotation of the mirror. In Fig.
2.35, I M is an incident ray, M R a
reflected ray. If the mirror be turned
into the position A'B', the reflected
ray is now M R' ; the reflected ray
has moved through the angle R M R',
which is equal to twice the angle
A M A'. This fact suggests an invalu-
able method of making minute motions
apparent. The reflected ray itself serving as a weightless index-
pointer of any desired length, and capable of magnifying motions to

INTENSITY OF REFLECTED LIGHT. 333

any desired extent. For example, the almost imperceptible motion
of the pulse may be made visible to a large audience in the follow-
ing manner : Lay (or
stick with wax) a tiny K
mirror upon the throb-
bing part of the wrist.
In a darkened room
project from a lantern
(or porte-lumifere) a
small beam of light

obliquely upon the ^ig 236

mirror M (Fig. 236),

and let the reflected beam strike the ceiling above. The spot of light
on the screen will move several inches with each pulsation.

294. Intensity of reflected light. — The intensity of reflected
light increases with the polish of the reflecting surface and
with the obliquity of the incident rays. It also depends
largely upon the nature of the medium from which it is
reflected. For example, at perpendicular incidence, water
reflects about the fiftieth part of the incident light while
mercury reflects about two-thirds ; but at an incidence of
89|-° each reflects about 72 per cent of the incident light.
The varnished surfaces of furniture appear much brighter
when viewed obliquely than when seen by light from a win-
dow reflected less obliquely. Light reflected from the surface
of a pond just before the sun sets is much more dazzling than
at noon when the sun is overhead. This is due in part to
the fact that we are in a suitable position to observe it.

295. Diffused light.

Experiment 2. — Introduce a small beam of light into a darkened room,
by means of a porte-lumiere, and place in its path a mirror. The light is
reflected in a definite direction. If the eye be placed so as to receive the
reflected light, it will se'e, not the mirror, but the image of the sun, and
the light will be painfully intense. Substitute for the mirror a piece of
unglazed paper. The light is not reflected by the paper in any definite
direction, but is scattered in every direction, illuminating objects in the

334

ETHER DYNAMICS.

vicinity and rendering them visible. Looking at the paper, you see, not
an image of the sun, hut the paper, and you may see it equally well in
all directions.

The dull surface of the paper receives light in a definite
direction, but reflects it in every direction ; in other words,
it scatters or diffuses the light. The difference in the phe-
nomena in the two cases is caused by the difference in the
smoothness of the two reflecting surfaces. AB (Fig. 237)
represents a smooth surface, like that of glass, which reflects
nearly all the rays of light in the same direction, because
nearly all the points of reflection are in the same plane.
C D represents a surface of paper having the roughness of
its surface greatly exaggerated. The various points of re-

FlG. 237.

flection are turned in every possible direction ; consequently,
light is reflected in every direction. Thus, the dull surfaces
of various objects around us reflect light in all directions,
and are consequently visible from every side. Objects ren-
dered visible by reflected light are said to be illum.inated.

By means of regularly reflected light we see images of
objects in mirrors, but only from definite positions, i.e. in
definite directions ; by means of diffused light we see the
object itself from every direction. Whether we see the image
of the source of the light (the eye being situated so as to
receive the regularly reflected light), or the object on which
the light falls, or both at the same time, depends largely
upon the degree of smoothness possessed by the surface that
reflects the light. Polished metals are the best mirrors.
Surfaces of liquids at rest are excellent mirrors. It is some-

REFLECTION FROM PLANE MIRRORS.

335

Fig. 238.

times difficult to see a smooth, surface of a pond surrounded
by trees and overhung by clouds, as the eye is occupied by
the reflected images of these objects ; but a faint breath of
wind, slightly rippling the surface, will reveal the water.

296. Reflection fi^om jplane mirrors ; virtual images. — MM
(Fig. 238) represents a plane mirror, and AB a pencil of
divergent rays proceeding from the
point A of an object A H. By erect-
ing perpendiculars at the points of
incidence, or the points where these
rays strike the mirror, and making
the angles of reflection equal to the
angles of incidence, the paths B C and
E C of the reflected rays are found.

Every visible point of an object
sends a cone of rays to the eye. The
point always appears at the place
whence these rays seem to emerge, i.e. at the real apex of
the cone. If the direction of these rays be changed by re-
flection, or in any other manner, the point will appear to be
in the direction of the rays as they enter the eye ; thus the
point A appears to lie in the direction C D ; and the point H,
in the direction G N. The exact location of these points may
be found by continuing the rays CB and CE behind the
mirror, till they meet at the points D and N. Thus, the
pencils E C and B C appear to emanate from the points N
and D ; and the whole body of light-waves received by the
eye seems to come from an apparent object ND behind the
mirror. This apparent object is called an image. An image
is a point or a series of points from which a diverging pencil
of rays comes or appears to come. As of course no real
image can be formed back of a mirror, such an image is
called a virtual or an imaginary image. It will be seen, by
construction, that an image in a plane mirror appears as far

336 ETHER DYNAMICS.

behind the mirror as the object is in front of it, and is of the
same size and shape as the object:

It appears from the above diagram that divergent incident
rays remain divergent after reflection from a plane mirror.
In a like manner the student may construct diagrams, and
show that parallel incident rays are parallel after reflection,
and convergent incident rays are convergent after reflection,
i.e. reflection from, a plane mirror does not change the angle
between rays.

297. Reversion of images. — When we look at our own
faces in a mirror we discover a lateral reversion. The right
cheek is the left cheek in the image ; the hair parted on the
left is parted on the right in the image.

If the mirror be vertical, objects appear in their proper
relations to the horizon ; but, if the mirror have any other
position, objects assume unnatural postures. Thus, turn this
book so that the mirror M M (Fig. 238) may represent a hori-
zontal mirror, and AH a vertical object above it, and it will
be seen that the image appears inverted. To verify this,
place a mirror in a horizontal position, and set on it a goblet
of water. The image of the goblet will appear upside down.
In a mirror inclined at an angle of 45° to the horizon, the
image of an erect object appears horizontal, while the image
of a horizontal object appears erect.

298. Multiple reflection; images of images.

"When light is reflected successively from two plane mirrors, the
image in the first becomes the object for the second mirror, and
the second image is found in precisely the same manner as the first
one. Again, the second image serves as an object for a third image,
and so on. If the two mirrors be parallel, as A and B (Fig. 239),
the series of images, theoretically infinite in number, is formed on
a common straight line normal to the mirrors and at regularly in-
creasing distances from the mirrors. Thus a' is the primary image
of the object a in mirror A (to avoid confusion, a pencil from only
one point o is drawn). The light reflected at c enters the eye as

MULTIPLE REFLECTION.

337

though it came from o\ Other rays reflected from A at e diverge
as though they emanated from a', and are reflected from B at e\
and may be regarded as proceeding from a real object at a', whose
image is 6, as far back of B as a' is in front of B. The light re-
flected from B to A again diverges as though it really came from 6,
and h being regarded as a real object, its image would be formed
at a'', and the pencil which enters the eye seems to proceed from

o''% having been reflected at e'' as though it came from o'\ The
pencil which would enter the eye from a third image at the left of
a" may be traced through all its reflections in like manner. As
some light is lost at each reflection, the images decrease in bright-
ness as they recede.

A kaleidoscope is constructed on the principle of multiple re-
flection. It consists of a tube containing three mirrors placed at
angles of 60°. Pieces of colored glass, free to move at one end of
the tube, are seen through an eye-piece
at the opposite end of the tube, multi-
plied by repeated reflections.

Multiplied images of a small, bright
object, as of a candle flame (Fig. 240),
often seen in a glass mirror, are pro-
duced by repeated reflections between
the anterior surface and the silvered
posterior surface of the mirror. At
each internal impact on the first sur-
face some light escapes, and shows us
an image, while another portion is re-
flected to the back, and thence forward
again, showing another image, and so on. Fig. 240.

Fig. 241.

ETHER DYNAMICS.

299. Reflection from concave mirrors. — ^Let MM' (Fig. 241)
represent a section of a concave spherical mirror, which may
be regarded as a small part of a hollow spherical shell having

a polished interior
surface. The distance
MM' is called the di-
ameter of the mirror.
C is the center of the
sphere, and is called
the center of curvature.
G is the vertex of the
mirror. A straight line DG drawn through the center of
curvature and the vertex is called the jprinci'pal axis of the
mirror. A concave mirror may be considered as made up of
an infinite number of small plane surfaces. All radii of the
mirror, as CA, CG, and CB, are perpendicular to the small
planes which they strike. If C be a luminous point, it is
evident that all light-waves emanating from this point, and
striking the mirror, will be reflected to their source at C.

Let E be any luminous point in front of a concave mirror.
To find the direction that rays emanating from this point
take after reflection, draw any two lines from this point, as
EA and EB^ representing two of the infinite number of rays
composing the divergent pencil that^trikes the mirror. ISText,
draw radii to the points of incidence A and B, and draw the
lines AF and BF, making the angles of reflection equal to
the angles of incidence. Place arrow-heads on the lines rep-
resenting rays to indicate the direction of the motion. The
lines AF and BF represent the direction of the rays after
reflection.

It will be seen that the rays after reflection are convergent,
and meet at the point F, called the focus. This point is the
focus of all reflected rays that emanate from the point E. It
is obvious that if F were the luminous point, the lines AE

REFLECTION FROM CONCAVE MIRRORS. 339

and B E would represent the reflected rays, and E would be
the focus of these rays. Since the relation between the two
points is such that light-waves emanating from either one are
brought by reflection to a focus at the other, these points are
called conjugate foci. Conjugate foci are two points so related
that the image of either is formed at the other. The rays EA
and E B, emanating from E, are less divergent than rays F A
and FB, emanating from a point F less distant from the
mirror, and striking the same points. Bays emanating from
D, and striking the same points A and B, will be still less
divergent ; and if the point D were removed to a distance of
many miles, the rays incident at these points would be very
nearly parallel. Hence rays may be regarded as practically
parallel when their source is at a very great distance, e.g. the
sun's rays. If a sunbeam, consisting of a bundle of parallel
rays, as E A, D G, and HB (Fig. 242), strike a concave mirror
in a direction parallel with its principal axis,
these rays become convergent by reflection,
and meet at a point (F) in the principal
axis. This point, called the principal focus,
is about halfway between the center of curva-
ture and the vertex of the mirror.

On the other hand, it is obvious that di-
vergent rays emanating from the principal focus of a concave^
mirror become parallel by reflection.

If a small piece of paper be placed at the principal focus
of a concave mirror, and the mirror be exposed to the parallel
rays of the sun, the paper will quickly burn.

Construct a diagram, and show that rays proceeding from a
point between the principal focus and the mirror are divergent
after reflection, but less divergent than the incident rays. On
reversing the direction of the rays, the same diagram will
show that convergent rays are rendered more convergent by
reflection from concave mirrors.

A
G

\^^ ' H
, 11

Fig. 242.

340

ETHER DYNAMICS.

The general effect of a concave mirror is to inci^ease the con-
vergence or to decrease the divergence of incident rays.

300. Spherical aberration of mirrors.

The statement that parallel rays after reflection from a concave
mirror meet at the principal focus is only approximately true. It
is strictly true only of parabolic mirrors such as are used in the
head-lights of locomotives. Consequently parabolic mirrors are
used when it is desired to bring the plane-fronted light-waves of a
distant star accurately to a focus, or to change a divergent pencil
to a parallel beam ; in the latter case the source of light is placed at
the focus of the paraboloid. In spherical mirrors when the pencil
is broad, the outside rays or those which are incident upon the
mirror farthest from its vertex are brought to a focus nearer the
mirror than the inner rays ; consequently the image furnished by
a luminous point is a circle brightest toward its center. This phe-
nomenon is called the spherical aberration of a mirror. It renders
the definition of the images of objects, especially of broad objects,

very bad. In conse-
quence of this it is often
necessary to cut off the
outside rays by a dia-
I)hragm, which improves
the definition at the ex-
pense of the brightness
of the image.

By constructing a
number of rays (Fig.
243) we may show that
all rays after reflection
are tangent to a charac-
teristic curve called a
caustic. The light emit-
ted from a single point, as A, is spread over the surface produced
.by the revolution of the line ST about the axis AM. This curve
formed in milk by reflection from the interior surface of a bright
tin pail is commonly called " the cow's foot."

301. Formation of images. — To determine the position
and kind of images formed in concave mirrors of objects

Fig. 243.

FORMATION OF IMAGES.

341

Fig. 244,

placed in front of them, proceed as follows : Locate the ob-
ject, as DE (Fig. 244). Draw lines, E A and DB, from the
extremities of the object through
the center of curvature of the mir-
ror, to meet the mirror. These lines
are called secondary axes. Incident
rays along these lines will return
by the same paths after reflection.
Draw another line from D to any
point in the mirror, e.g. to F, to rep-
resent another of the infinite num-
ber of rays emanating from D. Make the angle of reflection
CFD' equal to the angle of incidence CFD, and the reflected
ray will intersect the secondary axis D B at the point D'.
This point is the conjugate focus of all rays proceeding from
D. Consequently, an image of the point D is formed at D'.
This image is called a real image, because rays actually meet
at this point. In a similar manner, find the point E', the
conjugate focus of the point E. The images of intermediate

points between D and E lie
between the points D' and E' ;
and, consequently, the image
of the object lies between
those points as extremities.

If, for the second ray to be
drawn from any point, we
select that ray which is par-
allel with the principal axis,
as AG (Fig. 245), it will not be necessary to measure angles.
For this ray, after reflection, must pass through the principal
focus F ; and consequently the conjugate focus A' is easily
found, and so for the point B' and intermediate points. Both
methods of constructing images should be practiced by the
pupil.

Fig. 245.

342

ETHER DYNAMICS.

Fig. 246.

It thus appears that an image of an object placed beyond the
center of curvature of a concave TYiirror is real, inverted, smaller
than the object, and located between the center of curvature and
the principal focus of the mirror. An eye placed in a suitable
position to receive the light, as at H (Fig. 246), will receive

the same impression from the
reflected rays as if the image
E'D' were a real object. For
a cone of rays originally ema-
nates from (say) the point D of
the object, but it enters the
eye as if emanating from D',
and consequently appears to
originate from the latter point. A person standing in front
of such a mirror, at a distance greater than its radius of
curvature, will see an image- of himself suspended, as it were,
in mid-air. Or, if in a darkened room an illuminated object
be placed in front of the mirror, and a small oiled-paper screen
be placed where the image is formed, a large audience may
see the image projected upon the screen.

If E'D' (Fig. 246) be taken as the object, then the direc-
tion of the light in the diagram will be reversed, and ED
will represent the image.
Hence, the image of an
object placed between the
principal focus and the
center of curvature is also
real and inverted, hut
larger .than the object, and
located beyond the center
of curvature. The image in this case may be projected upon
a screen, but it will not be so bright as in the former case,
because the light is spread over a larger surface.

Construct an image of an object placed between the principal

Fig. 247.

FORMATION OF IMAGES.

343

;-c

Fig. 248.

focus and the mirror, as in Fig. 247. It will be seen in

this case that a pencil of rays proceeding from any point of

an object, e.(j. D, has no actual focus, but

appears to proceed from a virtual focus

D', back of the mirror, and so with other

points, as E. The image of an object

placed between the priiicipal focus aiid

the mirror is virtual, erect, larger than

the object, and bach of the mirror.

The diagram in Fig. 248 suggests the
method of finding the disposition of a pencil of rays emanating
from any point {e.g. A) after reflection from a convex mirror.
Construct an image of an object placed in front of a convex
mirror.

302. Illustrative experiments.

Experiment 3. — Hold some object, e.g. a rose, as ah (Fig. 249), a few
feet in front of a concave mirror. Looking in the direction of the axis

of the mirror you see a
small inverted image,
AB, of the object, be-
tween the center of cur-
vature, C, of the mirror
and its principal focus,

Evidently if AB rep-
resent an object placed
between the principal
focus and the center of
curvature, then ah will
represent the image of
the object.
Experiment 4- — Place a candle in an otherwise dark room 20 feet
from the mirror, catch the focused light-waves upon a paper screen, and
show that the focus is about half-way between the vertex and the center
of curvature of the mirror.

Experiment 5. — Advance the distant candle flame toward the mirror,
moving it up and down. (1) Show that the focus advances to meet the

Fig. 249.

344 ETHER DYNAMICS.

flame, and that when the flame is raised the focus is depressed, and the
converse. (2) Show that when the flame is at the center of curvature,
the focus is also there. (3) Show that when the flame is between the
center of curvature and the principal focus, tlie focus of the flame is
farther away than the center of curvature. (4) Show that when the
flame is at the principal focus, the reflected rays are parallel, or the focus
is at an infinite distance. (5) Show that when the flame is still nearer,
the reflected rays diverge and appear to come from a point behind the
mirror. (6) Notice that in all cases except the last the images are real
and inverted, and that in all cases where a real image is formed, the'
flame and the image may change places.

Experiment 6. — Form a real image of the flame between yourself and
the mirror; view the image through a convex lens (§ 318); show that
the image can be magnified by a convex lens, and thereby illustrate the
principle of the astronomical reflecting telescope (§ 388).

When light emitted by a luminous point at a distance / is reflected by
a concave spherical mirror, it is reflected back to an approximate focus
at a distance /'. The relation between the distance of the source /, the
distance of the focus /', and the radius of the mirror r, is expressed by
the following simple formula ^ :

,., yt

_2_
r

From this

equation

we get

(2) r=-^

which gives the distance of the image from the mirror in terms of / and r.

2
(a) Since the sum of the reciprocals of / and/' is a constant, -, it fol-

lows that as/ increases/' decreases, and when/ becomes infinite/' = -•

Hence parallel rays {i.e. rays from an infinitely distant source) come to a
focus at a point half way between a mirror and its center of curvature.

When/ = -, /'= CO ; i.e. rays emanating from the principal focus become

parallel, and the waves plane-fronted.

(6) When / decreases /' increases, i.e. the object and image approach

2 2
each other. When / equals /', ^= -, i-e. object and image coincide at

the center of curvature.

1 See Ganot, p. 434 ; Barker, p. 416.

FORMATION OF IMAGES. 345

r 1 2

negative, and the image is behind the mirror, and hence virtual. Dis-
tances in front of the mirror are considered positive, and those back of
the mirror negative.

Questions.

1. When an object is located at a distance from a concave mirror
equal to its radius, will any image be formed ? Why ?

2. What is the effect of placing the object at the principal focus?
Why?

3. a. When is the real image formed by a concave mirror smaller
than the object ? b. When is it larger ?

4. a. When is the image formed by a concave mirror real ? b. When
is it virtual ?

5. a. Is the image of an object formed by a convex mirror real or
virtual ? b. Is it larger or smaller than the object ? c. Is it erect or
inverted ?

6. Is the general effect of a convex mirror to collect or to scatter rays ?

7. The radius of a concave spherical mirror is 20 inches. Determine
the conjugate focus for a point on the principal axis 15 inches from the
mirror.

8. Why do images formed by a surface of v^ater appear inverted ?

9. a. What kind of wave-front has a beam of parallel rays ? b. What
change in front of such a beam occurs when it strikes each of the follow-
ing mirrors : viz. a plane, a concave, and a convex mirror ?

10. Where is the conjugate focus of light emanating from each of the
following points : a. the center of curvature of a concave spherical
mirror ; b. a point on the principal axis at an infinite distance from the
mirror ; c. a point on the principal axis beyond the center of curvature
and at a finite distance from the mirror ; d. a point on the principal axis
between the center of curvature and the principal focus ; e. a point be-
tween the principal focus and the vertex ?

11. How could you find the radius of curvature of a concave spherical
mirror by optical means alone ?

346

ETHER DYNAMICS.

Section VI.

REFRACTION.

303. Introductory experiments.

Experiment 1. — Into a darkened room admit a sunbeam so that its

rays may fall obliquely on the bottom of the basin (Fig. 250), and note

the place on the bottom where the edge of the shadow D E cast by the

side of the basin D C meets the bottom
at E. Then, without moving the basin,
fill it evenly full with water sliglitly
clouded with milk or with a few drops
of a solution of mastic in alcohol. It
will be found that the edge of the shadow
has moved from D E to D F, and meets
the bottom at F. Beat a blackboard
rubber, and create a cloud of dust in
the path of the beam in the air, and you
will discover that the rays GD that
graze the edge of the basin at D become
bent at the point where they enter the

water, and now move in the bent line G D F, instead of, as formerly, in

the straight line GDE. The path of the line in the water is now nearer

to the vertical side D C ; in other words, this part of the beam is more

nearly vertical tha^n before.

Experiment 2. — Place a coin (A, Fig. 251) on the bottom of an empty

basin, so that, as you look through a small hole in a card B C over the

edge of the vessel, the coin is just out of sight.

Then, without moving the card or basin, fill the

latter with water. Now, on looking through

the aperture in the card, the coin is visible. H

The beam AE, which formerly moved in the

straight line AD, is now bent at E, where it

leaves the water, and, passing through the aper- q

ture in the card, enters the eye. Observe that

as the beam passes from the water into the air it

is turned farther from a vertical line E F ; in other words, the beam is

farther from the vertical than before.

Fig. 251.

CAUSE OF REFRACTION. 347

Experiment 3. — From the same position as in the last experiment,
direct the eye to the point G in the basin filled with water. Reach your
hand around the basin, and place your finger where that point appears
to be. On examination, it will be found that your finger is considerably
above the bottom. Hence, the effect of the bending of rays, as they pass
obliquely out of water, is to cause the bottom to appear more elevated than
it really is; in other words, to cause the water to appear shallower than
it is.

Experiment 4. — Thrust a pencil obliquely into water ; it will appear
shortened, and bent at the surface of the water, and the immersed portion
will appear elevated.

Experiment 5. — Place a piece of wire (Fig. 252) vertically
in front of the eye, and hold a narrow strip of thick plate glass
horizontally across the wire, so that the light-waves from the ' |
wire may pass obliquely through the glass to the eye. The wire
will appear to be broken at the two edges of the glass, and the
intervening section will appear to the right or left according
to the inclination of the glass ; but if the glass be not inclined to the one
side or the other, the wire does not appear broken.

When a ray of light passes from one medium into another
of different density, it is bent or refracted at the interface
between the two mediums unless it meet this plane perpen-
dicularly. In the latter case there is no refraction. If it
pass into an optically denser medium, it is refracted toward
the perpendicular to this plane ; if into a rarer medium, it is
refracted from the perpendicular. It is not universally true
that the denser mediums are the more highly refracting.
The refractive power of water is less than that of alcohol or
oil of turpentine. A substance which has a higher refractive
powder than another is said to be oi)ticallij denser.

The angle GD (Fig. 250) is called the angle of iiicidence ;
EDN, the angle of refraction ; and EDF, the angle of devia-
tio7i.

304. Cause of refraction. — Foucault and others have
proved by careful experiments that the speed of light is
much less in water than in air. It is less in glass than in

348

ETHER DYNAMICS.

water, and much less in diamond than in glass. Every trans-
parent substance has its own rate of transmission. It would
seem that there is an interaction between the ether and the
molecules of matter such that in different mediums the ether-
waves are unequally retarded. T
Let the series of parallel lines AB (Fig. 253) represent a

series of wave-fronts leaving an
object C, and passing through a
rectangular piece of glass DE,
and constituting a beam. Every
point in a wave-front moves with
equal velocity as long as it trav-
erses the same medium ; but the
point a of a given wave ab enters
the glass first, and its velocity is
impeded, while the point b re-
tains its original velocity; so
that, while the point a moves to
a', b moves to b', and the result is that the wave-front assumes
a new direction (very much in the same manner as a line of
soldiers executes a wheel), and a ray or a line drawn perpen-
dicularly through the series of waves is turned out of its
original direction on entering the glass. Again, the extremity
c of a given wave-front cd first emerges from the glass, when
its velocity is immediately quickened ; so that, while d ad-
vances to d', c advances to c', and the direction of the ray is
again changed. The direction of the ray after emerging
from the glass is parallel to its direction before entering it,
but it has suffered a lateral displacement.

It is evident that if the ray enter the new medium in a
direction perpendicular to its surface, i.e. with its wave-
front parallel to this surface, all parts of the wave-front
will be retarded simultaneously and no refraction will take
place.

Fig. 253.

THE WAVE-THEORY APPLIED TO REFRACTION. 349

305. The ivave-theory ajjplied to refraction.

Let K A, ND, etc. (Fig. 254), be parallel rays of a beam of light
falling on a plane refracting surface AD. Let KLMN denote the
plane front of a wave. The wave reaches the refracting surface
first at A, and then
successively at other
points in A B C D.
As soon as the wave
reaches A, that point
becomes the origin
of an undulation
in the ether which
spreads out in all
directions in the me-
dium in the form of
a sphere having its
center at A. The
speed with which
motion is propagated

in the new medium we suppose to be less than that in the first
medium. Describe a circle with the center A, and with a radius
equal to the distance a wave would move in the new medium in the
same time as it would describe the excess of N D over K A in the
first medium. Let circles be described from B, C, and other points
of AD, according to the same law. Then a straight line a&cD
touching all these circles represents a plane refracted wave. It can
be demonstrated that the sine (see § 307) of the angle K A G bears
the same ratio to the sine of the angle a AH as the speed of light
in the first medium bears to the speed in the new medium, i. e. as
the excess of N D over K A bears to A a.

306. Failure of the emission theory to account for the refrac-
tion of light.

Fig. 254.

To explain refraction from a rare to a denser medium according
to the emission theory of light it is necessary to assume that when
a light particle shot from a luminous body comes within a very
small distance of the surface of separation between two mediums,
it begins to be attracted towards the surface so that its component

8:50

ETHER DYNAMICS.

velocity perpendicular to the surface gradually increases till it
reaches a limited distance on the other side of the surface. That is,
the speed of light should be by the theory greater in dense than in
rarer mediums ; whereas the reverse is found to be true. Hence
the failure of the emission theory to account for the phenomenon
of refraction.

307. Index of refraction. — The deviation of light-waves in
passing from one medium into another, depends upon the
mediums and the angle of incidence. It diminishes as the
angle of incidence diminishes, and is zero when the incident
ray is normal. It is highly important, knowing the angle of
incidence, to be able to determine the direction which a ray

will take on entering a new
medium. Describe a circle
around the point of inci-
dence A (Fig. 2ob) as a
center ; through the same
point draw IH perpendicu-
lar to the surfaces of the
two mediums, and to this
line drop perpendiculars
B D and C E from the points
where the circle cuts the
ray in the two mediums.
Then suppose that the per-
pendicular B D is j% of the radius AB ; now this fraction
y8g. is called (in trigonometry) the sine of the angle DAB.
Hence, -^^ is the sine of the angle of incidence. Again, if we
suppose that the perpendicular CE is f^ of the radius, then
the fraction -^^ is the sine of the angle of refraction. The
sines of the two angles are to each other as y8_ ; _6_^ or as
4:3. The quotient (in this case 1 = 1.33+) obtained by
dividing the sine of the angle of incidence by the sine of
the angle of refraction, generally expressed in the form of a

Fig. 255.

351

decimal fraction, is called the iiidex of refraction. It can be
proved to be the ratio of the velocity of the incident to that of
the refracted light-waves.

308. SnelVs ^^Law of Sines. ^^ — We have found that a ray
of light in passing obliquely from a medium into another of
different density suffers refraction, and the greater the angle
of incidence the greater the deflection. Snell in 1621 dis-
covered the law which governs these variable angles of de-
flection. It is called the "law of sines": ^^ The incident and
refracted rays are in the same plane with the normal to the
surface; they lie on opposite sides of it, and the sines of their
inclinations hear a constant ratio to each other." The incident
ray may be more or less oblique, still the index of refraction
remains the same.

309. Indices of refraction. — The index of refraction for
light-waves in passing from air into water is approximately f ,
and from air into glass f ; of course, if the order be reversed,
the reciprocal of these fractions must be taken as the indices ;
e.g. from water into air, the index is |; from glass into air, f.
When a ray passes from a vacuum into a medium, the re-
fractive index is greater than unity, and is called the absolute
index of refraction. The relative index of refraction, from any
medium A into another B, is found hy dividing the absolute
index of B by the absolute index of A.

It will be shown later that the refractive index varies with
wave-length. The following table is intended to represent
mean indices for light-waves : —

Table of Absolute Indices.

Lead chromate 2.97

Diamond (about) 2.5

Carbon disulphide 1.64

Flint glass (about) 1.61

Agate 1.54

Canada balsam 1.53

Crown glass (about) . . . 1.53

Spirits of turpentine . . . 1.48

Alcohol 1.37

Humors of the eye (about) . 1.35

Pure water . . . . . . . 1.33

Air at 0° C. and 760"™ pressure 1.000294

352

ETHER DYNAMICS.

310. Given the direction of the incident ray and the re-
fractive index, to determine the direction of the refracted ray. —
Let L (Fig. 256) be the incident ray ; draw a circle with

the point of incidence, 0, as a center.
Divide C by the refractive index, and
set off the quotient, D, on the other
side of 0. Draw DB perpendicular to
the surface at D, meeting the circum-
ference at B; then B is the direction
of the refracted ray required. For it is

Fig. 256. apparent that — - is the same as the

ratio of the sines, and this ratio is by construction equal to
the refractive index.

311. Some phenomena of refraction. Refraction into a
rarer medium. — A stick partly immersed in water appears
to be bent upwards and shortened unless its position is
vertical, when the part
immersed appears sim-
ply shortened. Fig.
257 explains this. The
dotted lines represent
the real position of the
submerged part of the
stick, and dotted lines
diverging from a point
at the bottom of the
stick show the course
of the rays which reach
the eye from that point. On reaching the surface they are
bent from the perpendicular, and the bottom of the stick is
seen in the direction from which the rays actually enter the eye.

Viewed obliquely, the depth of water cannot appear greater
than I its real depth. Hence the shoaling effect of still

Fig. 257.

CRITICAL ANGLE.

353

water in which the bottom is visible. To an eye under
water the surface must appear at least | of its real dis-
tance.

312. Critical angle; total reflection. — Let S S' (Fig. 258)
represent the boundary surface between two mediums, and
A and B incident rays in the more refractive medium
(e.g. glass) ; then D and E may represent the same rays
respectively after they enter the less refractive medium (e.g.
air). It will be
seen that, as the
angle of incidence
is increased, the re-
fracted ray rapidly
approaches the sur- _
face S. Now, |
there must be an |
angle of incidence 1
(e.^. COM) such that |
the angle of refrac- I
tion will be 90°; in [
this case the inci-
dent ray CO, after
refraction, will just graze the surface S. This angle (COM)
is called the critical or limiting angle. Any incident ray, mak-
ing a larger angle with the normal than the critical angle, as
LO, cannot emerge from the medium, and consequently is
not refracted. Experiment shows that all such rays undergo
internal reflection ; e.g. the ray L is reflected in the direc-
tion ON. Reflection in this case is perfect, and hence is
called total reflection. Total reflection occurs when rays in the
more refractive medium are incident at an angle greater than
the critical angle.

Surfaces of transparent mediums, under these circum-
stances, constitute the best mirrors possible. The critical

Fig. 258.

354

ETHER DYNAMICS.

angle diminishes as the refractive index ^ increases. For
water it is about 48|-° ; for flint glass, 38° 41'; and for the
diamond, 23° 41'. Light-waves cannot, therefore, pass out of
water into air with a greater angle of incidence than 48^°.
The brilliancy of gems, particularly the diamond, is due in
part to their extraordinary power of reflection, arising from
their large indices of refraction.

313. Illustrations of refraction and total reflection.

Experiment 6. — Observe the image of a candle flame reflected by the
surface of water in a glass beaker, as in Fig. 259.

Experiment 7. — Thrust the closed end of a glass test-tube (Fig. 260)
into water, and incline the tube. Look down upon the immersed part

Fig. 259.

Fig. 260.

of the tube, and its upper surface will look like burnished silver, or as if
the tube contained mercury. Fill the test-tube with water, and immerse
as before ; the total reflection which before occurred at the surface of the
air in the submerged tube now disappears. Explain.

1 It must be evident on inspection of Fig. 258 that a ray traveling in the direc-
tion SO will be refracted in the direction OC. The angle SOK is a right angle,
and the sine of a right angle = 1. Therefore the index of refraction of the medium

^=— : Thus, to get the index of refraction of any substance it is only

sm crit. ang.

necessary to find the critical angle of the substance. This principle has been applied

by Kohlrausch in his total reflectometer, in determining indices of refraction of

crystals.

A LUMINOUS CASCADE.

355

Fig. 261.

A glass prism of 90° is often used as a reflector. Light

passes through the surface AB (Fig. 261) and meets the

surface AC at an angle of 45°, which is ^

greater than the critical angle for glass.

It is therefore totally reflected, and the

device is consequently more effective y

than an ordinary mirror.

314. A luminous cascade. — If water

be siphoned through a glass tube having

an open side tubule, a (Fig. 262), and

the tube below a be exposed to the

direct rays of the sun or placed in the path of a beam of

light from a lantern or porte-lumiere, the stream of water
mingled with air which enters the
tubule will appear like a '^stream
of living fire," and has received the
name of luininous cascade. This is
due to total reflection. The light in
passing through the water meets the
surface of the air-bubbles at angles
greater than the critical, and is
reflected from side to side all down
the stream.

Fig. 262.

Experiment 8. — Place uncolored glass beads, or glass broken into small
pieces in a test-tube. They appear not only white, becaiise of diffused
reflection, but quite opaque, because of refraction and internal reflection.
Pour some water into the tube, and it becomes somewhat translucent.
Substitute spirits of turpentine for the water, and the translucency is
increased.

By mixing a small quantity of carbon bisulphide with the turpentine,
or olive oil with oil of cassia, a liquid can be obtained whose refractive
index is about the same as that of glass, when the light will pass through
the liquid without obstruction, and the beads become transparent and
nearly invisible. The last illustration shows that one transparent body
within another can he seen only when their refractive indices differ. Place

356

ETHER DYNAMICS.

your eye on a level with the surface of a hot stove, and you may observe
a w&yy motion in the air, due to the mingling of currents of heated and

less refractive air with cooler and
more refractive air.

A ray of light from a heavenly
body A (Fig. 263) undergoes a series
of refractions as it reaches successive
strata of the atmosphere of constantly
increasmg density, and to an eye at
the earth's surface appears to come
from a point A' in the heavens. The
general effect of the atmosphere on
the path of light that traverses it is
such as to increase the apparent
altitude of the heavenly bodies. It
enables us to see a body (B) which
is actually below the horizon, and
prolongs the apparent stay of the sun, moon, and other heavenly bodies,
above the horizon, i Twilight is due both to refraction and reflection of
light by the atmosphere.

Fig. 2G3.

Exercises.

1. Draw a straight line to represent a surface of flint glass, and draw
another line meeting this obliquely to represent a ray of light passing
from a vacuum into this medium. Find the direction of the ray after it
enters the medium, employing the index as given in the above table.

2. a. Determine the relative index of refraction for light in passing
from water into diamond. 5. In passing from water into air.

3. How must one modify his aim in shooting or spearing fish from the
bank of a stream ?

4. A ray is incident on a surface of crown glass at an angle of 40° ;
find the angle of refraction.

5. Find the refractive index for light passing from water into crown
glass.

6. Does a star in the zenith appear to be where it really is ? Why ?

1 Under average conditions the refraction elevates a body at the horizon about 35',
so that both the sun and the moon in rising appear clear of the horizon while still
wholly below it. The amount of refraction varies sensibly with the temperature and
barometric pressure, increasing as the thermometer falls or as the barometer rises.

OPTICAL PRISMS.

357

Section VII.

PRISMS AND LENSES.

315. optical prisms. — An optical prism is a portion of a
transparent medium bounded by plane surfaces inclined to
each, other. Eig. 264 represents a transverse section of a
common form of prism. Let
AB be a ray of light incident
upon one of its surfaces. On
entering the prism it is re-
fracted toward the normal, and
takes the direction B C. On
emerging from the prism it is
again refracted, but now from

the normal in the direction C D. The object that emits the ray
will appear to be at F. Observe that the ray A B, at both refrac-
tions, is bent toward the thicker part, or base, of the prism.

316. Measuring iiidex of refraction.

The ray S (Fig. 265) strikes the face A C of a prism at tlie angle
of incidence i, and is refracted at the angle r ; sin i = n sin r, (1), n
being the index of refraction. It strikes the face A B at the angle

r\ and leaves the prism
A at the angle i' \ sin i' =

n sin r' (2). It may
be proved geometrically
that the angles r and r'
are together equal to the
angle of the prism A ;
r+r'= A (3). Also, if
the angle between the
incident ray SD pro-
duced and the deviated ray E S' be Y {i.e. the angle GO S', called the
angle of deviation) then i +V =Y-\- A(4) . From these four equations,
which involve n, together with any three of the six angles, z, i', r,
r^, A, V, we may determine for any given monochromatic light
(§ 357) the index of refraction n of the material of the prism.

Fig. 265.

358

ETHER DYNAMIC So

317. Minimum deviation.

Furthermore, it can be shown that when the prism is so placed
that i' becomes equal to z, the angle of deviation, V, of the ray has
its least value. Such a position is shown in Fig. 264, and is called
the position of minimum deviation. It is easily obtained in practice
by turning the prism until a certain position is obtained where the
beam of light, S^ comes to a standstill and begins to move back, no
matter which way the prism is rotated.

318. Lenses. — Any transparent medium bounded by sur-
faces of which, at least one is curved, is a lens.

Lenses are of two classes, converging and diverging, ac-
cording as they collect rays or cause them to diverge. Each
class comprises three kinds (Fig. 266) : —

1. Bi-convexi.

2. Plano-convex

3. Concavo-convex

(or meniscus)

Class I.

Converging, or
convex lenses,
thicker in the
middle than at
the edges.

Class II.

4. Bi-concave ] Diverging, or con-

5. Plano-concave I cave lenses, thin-

6. Convexo-con- [ ner in the middle

cave J than at the edges.

A straight line normal to both surfaces of a lens and pass-
ing through their centers of curvature, as A B, is called

its principal axis.
There is a point in ■
the principal axis
of every lens called
its optical cefiter.
This point is so
placed that a ray whose direction within the lens passes
through it suffers no angular deviation, but at most only a
slight lateral displacement. In lenses 1 and 4 it is half-way
between their respective curved surfaces.^ A ray drawn

1 If the two convex surfaces be of different curvature, the lens is called a
■" crossed lens."

2 In lens 2 the optical center is in its convex surface ; in lens 5 it is in its concave
surface ; in lenses 3 and 6 it is without the lens.

Fig. 266.

EFFECT OF LENSES.

359

through the optical center from any point of an object, as Ka
(Fig. 274j, is called the secondary axis of this point.

319. Ejfect of lenses, — We may, for convenience of illus-
tration, regard a biconvex lens as composed, approximately,
of a series of prisms of gradually increasing angles arranged
around an axis, as represented in section in Fig. 267. It
is apparent that the parallel rays
farthest from the principal axis, /i^.

meeting prisms of greater and /j\

greater angles of incidence, are
more deflected than those nearer the
axis ; and if the curvatures be
properly adjusted, all may be made
to converge to one point.

On the other hand if the lens be
thinnest at the center and gradually
increase in thickness outward, exact-
ly the opposite effect would be ex-
pected. Parallel incident rays, being
bent toward the thicker part of the
component prisms, would become
separated.

The general effect of all convex lenses is to cause transmitted
rays to converge ; that of concave lenses, to cause them to diverge.
Incident rays parallel to the principal axis of a convex lens

are brought to a focus F
(Fig. 268) at a point in the
principal axis. This point
is called the 'princi-pal focus,
i.e. it is the focus of incident
rays parallel to the principal
axis. It may be found by holding the lens so that the rays
of the sun may fall perpendicularly upon it, and then moving
a sheet of paper back and forth behind it until the image of

Fig. 267.

Fig. 268.

360

ETHER DYNAMICS.

Fig. 269.

the sun formed on the paper is brightest and smallest. Or,
in a room, it may be found approximately by holding a lens
at a considerable distance from a window, and regulating the
distance so that a distinct image of the window will be pro-

jected upon
the opposite
wall, as in
Pig. 269. The
focal length is
the distance
from the opti-
cal center of
the lens to the
center of the
image on the
paper. The

shorter the focal length the more powerful is the lens ;
that is, the more quickly are the parallel rays that traverse
different parts o-f the lens brought to cross one another.

If the paper be kept at the principal focus for a short time,
it will take fire. The reason is apparent why convex lenses are
sometimes called
"burning glasses."
A pencil of rays,
emitted from the
principal focus F
(Fig. 268) as a
luminous point, be-
comes parallel on
emerging from a convex lens. If the rays emanate from a
point nearer the lens, they diverge after egress, but the
divergence is less than before ; if from a point beyond the
principal focus, the rays are rendered convergent. A concave
lens causes parallel incident rays to diverge as if they came

Fig. 270.

CONJUGATE FOCI. 361

from a point, as F (Fig. 270). This point is therefore its
principal focus. It is, of course, a virtual focus.

Every lens has ^ principal focus ; it is the point to which
parallel rays are caused to converge, or from which, after
deflection, they appear to diverge, as the case may be.

320. Conjugate foci. — When a luminous point beyond the
principal focus, S (Fig. 271), sends rays to a convex lens, the
emergent rays
converge to an-
other point S' ;
rays sent from
S' to the lens
would converge ^^^- '^'^^•

to S. Two points thus related are called coiyjugate foci. The
fact that rays which emanate from one point are caused by
convex lenses to collect at one point, gives rise to real images,
as in the case of concave mirrors.

321. Laiv of converging lenses.

Lenses, like mirrors, have conjugate foci at distances p and p'
from the optical centers. In converging lenses the principal focal
distance and the distance of their conjugate foci (or distance of ob-
ject and image) are related according to the formula

V P' f

Hence the law of converging lenses : The reciprocal of the princi-
pal focal length is equal to the sum of the reciprocals of any two con-
jugate focal lengths.

When a pencil of light comes from an infinite distance (i.e. when
its rays are parallel), p = co ; then p' =/, and the rays converge at
the principal focus. Conversely, if a pencil come from the principal
focus, p =f; hence _2)'= co ; that is, no image is formed.

If the object (i.e. the source of light) be at a distance less than
infinity, but greater than 2/, the image is real, and is on the other
side of the lens at a distance greater than / and less than 2/ Con-
versely, if the object be at a distance greater than /, but less than
2/, the image is at a distance greater than 2/.

362 ETHER DYNAMICS.

If the object be at a distance 2/, the image is also at the distance
. 2/, and object and image are of equal size. This suggests a simple
way of finding /. Adjust an object, convex lens, and screen, so
that the image on the screen is equal in size to the object. Half
the distance of either the object or its image from the center of the
lens is the focal length of the lens.

322. Diverging lenses.

The formula for these is "> = 7 '

P V f

When p = CO , p' = — / (a virtual image at the principal focus i).
When _p = /, _p' = — CO (no image).

When p is of any value greater than /, and less than co , _p' Is
greater than /.

323. Images fanned. — Fairly distinct images of objects
may be formed through very small apertures (§ 283); but
owing to the small amount of light that passes through the
aperture, the images are very deficient in brilliancy. If the
aperture be enlarged, brilliancy is increased at the expense of
distinctness. A convex lens enables us to obtain both brilliancy
and distinctness at the same time.

Experiment 1. — By means of a porte-lumi^re, A (Fig. 272), introduce
a horizontal beam of light into a darkened room. In its path place some
object, as B, painted in transparent colors or photographed on glass.
(Transparent pictures are cheaply prepared by photographers for sunlight
and lime-light projections.) Beyond the object place a convex lens L,
and beyond the lens a screen S. The object being illuminated by the
beam of light, all the rays diverging from any point a are bent by the
lens so as to come together at the point a'. In like manner, all the rays
proceeding from c are brought to the same point c' ; and so also for all
intermediate points. Thus, out of the billions of rays emanating from
each of the millions of points on the object, those that reach the lens are
guided by it, each to its own appropriate point in the image. It is
evident that there must result an image both bright and distinct, pro-
vided the screen be suitably placed, i.e. at the place where the rays meet.

1 The negative sign refers to the direction in which p' is measured. Conjugate
foci of diverging lenses are on the same side of the lens.

IMAGES FORMED.

363

But if the screen be placed at S' or S", it is evident that a blurred image
will be formed. Instead of moving the screen back and forth, in order
to " focus " the rays properly, it is customary to move the lens.

Experiment 2. — Make a series of experiments similar to those with
the concave mirror. Ascertain the focal length of the convex lens.

\\)\\-\Vw\V\

Fig. 272.

Place the lens at a distance from a white wall about equal to its focal
length. Place a candle flame (better the flame of a fish-tail burner) at
such a distance the other side of the lens that it will produce a distinct
and well-defined image on the wall (Fig. 273). (1) Observe and note on

lilliiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiy

Fig. 273.

paper the size and kind of image. Advance the flame toward the lens,
regulating at the same time the distance between the lens and wall, so as
to preserve a distinctness of image. (2) Note the changes which the

364

ETHEH DYNAMICS.

image undergoes. (3) When the image and the flame become of the
same size, measure and note the distance of each from the lens. (4) Ad-
vance the flame still nearer, and note the changes in the image, until it is
impossible to obtain an image on the wall. Measure the distance of the
flame from the lens, and compare this distance with the focal length of
the lens. (5) Move the flame still nearer. Note whether the rays, after
emerging from the lens, are divergent or convergent. (6) See whether an
image and an object may change places. (7) Form images of the flame
on the wall at different distances from the lens ; measure the distances,
also the linear dimensions {e.g. the width, or the vertical hight) of the
images, and determine whether the linear dimensions of images are pro-
portional to their distances from the lens.

324. To construct the image formed by a convex lens. —
Given the lens L (Fig. 274), whose principal focus is at E,
and object AB in front of it ; any two of the many rays from

Fig. 274.

A will determine where its image a is formed. Two that can
be traced easily are, one along the secondary axis A a, and
one parallel to the principal axis A A': the latter will be
deviated so as to pass through the principal focus F, and
will afterward intersect the secondary axis at some point a ;
therefore this is the conjugate focus of A. Rays can be
similarly traced for B, and all intermediate points along the
arrow. Thus, a real inverted image is formed at a h.

The linear dimensions of an object and of its image formed
by a convex lens are jproportional to their resjicctive distances
from the center of the lens. The image is virtual or real, erect
or inverted, according as it is on the same side of the lens
with the object or on the op^josite side.

VIRTUAL IMAGES. 365

325. Virtual images. — Since rays that emanate from a
point nearer the lens than the principal focus diverge after
egress, it is evident that their focus must be virtual and on
the same side of the lens as the object. Hence, the image of
an object placed nearer the lens than the principal focus is
virtual^ magnified, and erect, as shown in Fig. 275. A convex
lens used in this manner is called a simple microscope.

326. Simple microscope. As its name implies, the micro-
scope is an instrument for viewing minute objects. The
simple microscope consists of a single converging lens so
placed that the object is between the principal focus and the
lens. It magnifies by increasing the visual angle.

A ,-'-'

-,B ^---^

Fig. 275.

The magnifying poiuer of the lens is simply the ratio
between the apparent diameter of the image and the diameter
of the object, e.g. A'B' : AB (Fig. 275), or it is the ratio
between the visual angles under which the eye would see
image and object, if both were placed at the distance of
distinct vision.^ If the lens be of short focus, as is usually
the case, the magnifying power is approximately the ratio of
the distance of distinct vision to the focal length. Thus a
lens of -J- in. focal length would magnify 20 to 24 diameters.

- 1 For normal eyes, an object to be seen most distinctly must be placed at a
distance of 10 to 12 inclies, bence this is regarded as the distance of distinct vision.

366

ETHER DYNAMICS.

327. Diverging lenses. — Since the effect of concave lenses
is to render transmitted rays divergent, pencils of rays
emitted from A and B (Fig. 276) diverge after refraction, as
if they came from A' and B', and the image will appear to be
at A' B'. Hence, images formed hy concave lenses are virtual,
erect, and smaller than the object.

Fig. 276.

328. Spherical aberration. — In all ordinary convex lenses
the curved surfaces are spherical, and the angles which inci-
dent rays make with the little plane surfaces of which we
may imagine the spherical surface to be made up, increase

Fig. 277.

rapidly toward the edge of the lens. Thus, while those rays
from a given point of an object which pass through the cen-
tral portion, as A (Fig. 277), meet approximately at the same
point F, those which pass through the marginal portion are
deflected so much that they cross the axis at nearer points,
e.g. at F' ; so a blurred image results. This wandering of the
rays from a single focus is called spherical aberration.

No lens with spherical surfaces can bring all the rays to
the same focus. The ev.il may be in a measure corrected by

SPHERICAL ABERRATION. 367

interposing a diaphragm D D' provided with a central aperture
smaller than the lens, so as to cut off those rays that pass
through the marginal part of the lens. But it can be wholly
corrected only by properly modifying the curvature of the
surfaces of the lens. A lens having surfaces thus modified is
said to be aplanatic.

Experiment 3. — (Illustrating spherical aberration.) Cut a cardboard
disk as large as the convex lens to be employed. Cut a ring of holes
near the circumference, and also a ring near the center. Support the
disk close to the lens, so as to cover one of its surfaces. Place the whole
in a beam from a porte-lumiere. Catch refracted beams on a screen.
Move the screen away from the lens. The beams through the outer ring
of spots are the first to cross one another and form an image. Further
away, the inner beams coincide, forming an image. The outer ones,
having crossed, form a ring of spots.

Qiiestions.

1. What must be the position of an object with reference to aeon-
verging lens, that its image may be real and magnified ?

2. A photographic transparency is placed between a porte-lumi6re and
a biconvex lens, 16 in. from the latter ; how many diameters is a distinct
image of the transparency multiplied on a screen 20 ft. distant ?

3. A transparency whose dimensions are 3 X 4 in. is placed 16 in. from
the lens ; at what distance from the lens must the screen be that it may
receive a distinct image of the transparency that shall cover a surface
3 X 4 ft. ?

4. What is the focal length of the lens used in the last case ?

5. With a converging lens the image of a candle is thrown on a screen
6 ft. from the candle, and the focal length of the lens is 16 in. ; find the
distances of the candle and of the screen from the lens. Ans. 4 ft. and 2 ft.

6. A luminous point is 3 in. from a convex lens having a focal length
of 5 in. ; find the position of the image.

7. If the candle and screen be 3 ft. apart, and the lens midway between
them, what is the focal length ?

8. Find the focal length of a lens which throws the image of an object
5 ft. distant on a screen 3 ft. distant.

9. A double concave lens having a focal length of 3 in. is held at a
distance of 2 in. from a small object ; find the position of the image.

368

ETHER DYNAMICS.

10. If an object be at twice tlie focal distance of a convex lens, how
will the length of the image compare with the length of the object ?

11. To an eye whose distance of distinct vision is 25 cm, how many-
diameters will a lens of 1 cm focus magnify ?

12. Show that a concave air lens in water has the same effect on inci-
dent light as a convex water lens in air.

Section VIII.

PKISMATIC ANALYSIS OF LIGHT. SPECTRUMS.

329. Analysis of light which produces the sensation of white.

JExperiment 1. — Place a disk with an adjustable slit in the aperture of
a porte-lumiere, so as to exclude from a darkened room all light-waves
except those which pass through the slit. Near the slit interpose a

Fig. 278.

double-convex lens of (say) 10-inch focus. A narrow sheet of light will
traverse the room and produce an image, AB (Fig. 278), of the slit on a
white screen placed in its path. Now place a glass prism C in the path
of the narrow sheet of light and near to the lens, with its edge vertical.

ANALYSIS OF LIGHT. 369

(1) The light now is not only turned from its former path, but that which
before was a narrow sheet, is, after emerging from the prism, spread out
fan-like into a wedge-shaped body, with its thickest part resting on the
screen. (2) The image, before only a narrow, vertical band, A B, is now
drawn out into a long horizontal ribbon DE. (3) The image, before
white, now presents all the colors of the rainbow, from red at one end
to violet at the other ; it passes gradually through all the gradations of
orange, yellow, green, blue, and violet. (The difference in deviation
between the red and the violet is purposely much exaggerated in the
figure.)

From this experiment Ave learn (1) that ivhite light is not
simple in its composition, hut the result of a mixture of colors}

(2) The colors of luhich white light is composed may be sepa-
rated hy refraction. (3) The separation is due to the different
degrees of deviation which colors undergo hy refraction. Eecl,
which is always least turned aside from a straight path, is
the least refrangible color. Then follow orange, yellow,
green, blue, and violet, in the order of their refrangibility.
The many-colored ribbon of light DE is called the solar spec-
trum? This separation of white light into its constituents is
called dispersion. The number of colors of which white light
is composed is really infinite, but we have names for only
seven of them ; viz. red^ orange^ yellow, green, cyan-hlue, ultra-
marine-hlue, and violet ; and these are called the primary or
prismatic colors. The names of the blues are derived from
the names of the pigments which most closely resemble them.

The spectrum may be projected upon a screen, or it may be
received directly by the eye, as in the two following experi-
ments : —

Experiment 2. — Upon a black cardboard A (Fig. 279) paste a strip
of white paper 3 cm long and 2 mm wide ; and place the prism and the
eye as in the figure. Now when a beam of white light from the strip is

1 Newton (1666) was the first to recognize the true import of this phenomenon, i.e.
to refer the colors to the heterogeneity of white light.

2 A succession of colors in the order of their refrangibility, obtained from any
source of light, is called a spectrin.

370

ETHER DYNAMICS.

refracted and dispersed by the prism and falls upon the retina of the eye,
you see, not the narrow white strip in its true position, but a spectrum
in the position A^ This experiment is performed in a lighted room.

Experiment 3. — Instead of a continuous white
strip, paste short strips of red, white, and blue,
end to end, on the black card, as represented in
Fig. 280. The spectrum of each color is given on
the right, the light portions representing the illu-
minated parts. It will be seen
that in the spectrum of the red,
the green, blue, and violet
portions are almost completely
dark, but there is a faint trace
of orange ; in the spectrum of
the blue, the red, orange, and
yellow are wanting, blue and
violet are present, and a small
quantity of green.

Fig. 279.

Fig. 2g

330. Synthesis of white light. — The composition of white
light has been ascertained by the process of analysis ; it can
be verified by synthesis ; i.e. the colors after dispersion may
be reunited, and the result of the reunion is white light.

Experiment 4. — Place a second prism (2) in such a position /^ that
light which has passed through one prism (1), and been refracted and
decomposed, may be refracted back, and the colors will be reblended,
and a white image of the slit will be restored on the screen.

Experiment 5. — Place a large convex lens, or a concave mirror
(better a concave cylindrical mirror), so as to receive the colors after dis-
persion by a prism, and bring the rays to a focus on a screen. The
image produced will be white.

Experiment 6. — Receive the spectrum on a common plane mirror,
and rapidly tip the mirror back and forth in small arcs, and the light
reflected by the mirror upon a screen will produce a white image on the
screen.

331. The rainbow. — The rainbow is a solar spectrum on a
grand scale. It is the result of refraction, total reflection,

CHROMATIC ABERRATION.

371

and dispersion, of sunlight by falling raindrops. Let spheres
1 and 2 (Fig. 281) represent drops at the extreme opposite
edges of the bow. The eye is in a position to receive, after
the dispersion and internal reflection of the light-waves within
drop 1, only the red waves ; consequently this part of the
bow appears red. So, likewise, from drop 2 the eye receives
only violet ; consequently this part of the bow appears violet.
In like manner, the intermediate colors of the bow are
sifted out.

Outside the primary bow a secondary/ bow is sometimes seen.
Drops 3 and 4 (Fig. 281) are supposed to be at the opposite

Fig. 281.

edges of the secondary bow. It will be seen that the light-
waves undergo two internal reflections within the drops
which produce this bow. The colors of this bow are in
reverse order to those of the primary bow, and less brilliant.
332. Chromatio aberration. — There is also in ordinary
convex lenses a serious defect, to which we have not before
referred, called chromatic aberration, the correction of which
has demanded the highest skill. The convex lens both refracts

372 ETHER DYNAMICS.

and disperses the light-waves that pass through it. The ten-
dency, of course, is to bring to a focus the more refrangible
rays, as the violet, much sooner than the less refrangible
rays, such as the red. The result is a disagreeable coloration
of the images that are formed by the lens, especially by those
portions of the light-waves that pass through the lens near
-.^^^^^^^ ^^^^^^^ its edges. This evil has been overcome very
^ .^-^r^r-r-^-,^^^ effectually by combining with the convex lens
a plano-concave lens. Now, if a crown-glass
jjiG. 282. convex lens be taken, a flint-glass concave

lens may be prepared that will correct the dispersion of the
former without neutralizing all its refraction.^ A compound
lens composed of these two lenses cemented together (Fig. 282)
constitutes what is called an achromatic lens.

333. Cause of color and dis2)ersio7i. — The color of light is
determined by vibration-frequency, or, in other words, by the
corresponding wave-length. The light-waves diminish in
length from the red to the violet. As pitch depends on
the frequency with which aerial waves strike the ear, so color
depends upon the frequency with which ether-waves strike
the eye. The difference between violet and red is a difference
analogous to the difference between a high note and a low
note on a piano.

The speed of propagation in a vacuum appears to be the
same for all wave-lengths. But in a refracting medium, the
short waves are more retarded than the longer ones, hence
they are more refracted. This is the cause of dispersion.
Each wave-length has its own refractive index, or, since
vibration-frequency corresponds to color, every simple color
has its special refractive index. Light composed of waves all
of the same (or nearly the same) length is called homogeneous
or monochromatic light. The yellow light emitted by the
flame of a Bunsen burner or alcohol lamp when common salt

1 The refractive and dispersive powers of the two lenses are not proportional.

CAUSE OF COLOR AND DISPERSION. 373

is sifted upon it is approximately monochromatic. Ordinary
white light is a mixture of long and short ether-waves.

From well-established data, determined by a variety of
methods, physicists have calculated the number of waves that
succeed one another for each of the several prismatic colors,
and the corresponding wave-lengths ; the following table con-
tains the results.^ The letters A, C, J), etc. refer to Fraun-
hofer's lines (see § 340).

Length of waves No. of waves

in millimeters. per second.

Dark red A 000760 395,000,000,000,000

Orange C .000656 458,000,000,000,000

Yellow D 000589 510,000,000,000,000

Green E 000527 570,000,000,000,000

C. Blue F 00048G 618,000,000,000,000

U. Blue G 000431 697,000,000,000,000

Violet H 000397 760,000,000,000,000

There is a limit to the sensibility of the eye as well as of
the ear. The limit in the number of vibrations appreciable
by the eye lies approximately within the range of numbers
given in the above table ; i.e. if the succession of waves be
much more or much less rapid than is indicated by these
numbers, the sensation of sight is not produced.

" Our knowledge of ether-waves is at present limited to
those which lie between 107 trillions and 40,000 trillions per
second — a range, in musical parlance, of about 8^ octaves "
(Langley). Of these our eyes are sensitive to scarcely one
octave.

It is evident that the frequency of the tvaves emitted hy a
luminous body, and consequently the color of the light emitted,

1 " That man should he able to measure, with certainty, such minute portions of
space and time, is not a little wonderful ; for, whatever theory of light we adopt, it
may he observed that these periods and these spaces have a real existence, being, in
fact, deduced by Newton himself from direct measurements, and involving nothing
hypothetical but the names here given them." — Sir John Herschel.

If science in the future shall be able to dispense with the ether of space, the
vibration periods and what corresponds to wave-lengths will necessarily remain.

374 ETHER DYNAMICS.

must depend on the rapidity of the vibratory jnotions of the
molecules of that body, i.e. upon its temperature. This has
been shown in a convincing manner* as follows : The temper-
ature of a platinum wire is slowly raised by passing a
gradually increasing current of electricity through it. At a
temperature of about 540° C. it begins to emit light ; and if
the light be analyzed by a prism, it is shown that only red
light is emitted. As the temperature rises, there will be
added to the red of the spectrum, first yellow, then green,
blue, and violet successively. When it reaches a white heat,
it emits all the prismatic colors. It is significant that a
white-hot body emits more red light than a red-hot body, and
likewise more light of every color than at any lower temper-
ature. The conclusion is, that a body which emits luhite light
sends forth simultaneously tuaves of a variety of lengths.

334. Continuous spectrums. — The spectrum produced by
the platinum is continuous ; that is, the band of light is
unbroken. If the spectrum be not complete, as when the
temperature is too low, it will begin with red, and be con-
tinuous as far as it goes. All luminous solids and liquids give
continuous spectrums.

A gas, kerosene, or candle flame does not give the spectrum
of a vapor, but gives that of the solid particles of carbon in a
state of incandescence ; hence the continuous spectrums which
these flames afford.

335. Spectroscopes. — Instruments for the observation of
spectrums are called spectroscopes. The essential part of the
apparatus is the "dispersion piece," which is either a prism
or a diffraction grating (see Fig. 302). Instead of looking at
the spectrum with the naked eye it is usually better to view
it through a small telescope, which serves to magnify it.
Fig. 283 represents the simplest form of the Kirchhoff and
Bunsen spectroscope. A flint glass prism receives light
through an adjustable slit at the end of a tube called the

SPECTROSCOPES. 375

collimator. At the opposite end of this tube is a converging
lens, and the slit is located at its principal focus so that rays
diverging from the slit are rendered parallel by the lens.

It is often necessary to have some means of determining
the positions of certain lines (to be described hereafter)

Fig. 283.

observed in the spectrum. The usual method is to have a
second tube, somewhat like the collimating tube, so placed
that the rays from a light (e.g. a candle flame as in Fig.
283) after passing through a transparent plate (inside the
tube) on which a fine scale is engraved, and through a lens,
by which they are made parallel, are reflected from the

376

ETHER DYNAMICS.

nearest face of the prism, and pass into the telescope along
with the beam of light under analysis. Thus the eye while
viewing the spectrum through the telescope sees also a
magnified image of the scale coinciding with the spectrum.

336. Direct -vision pocket spectroscope. — A small instru-
ment called a pocket spectroscope will answer fairly well for
experiments given in this book. This instrument contains
three or more prisms, A, B, and C (Fig. 284). The prisms
are enclosed in a brass tube, D, and this tube in another tube,
E. F is a convex lens, and G is an adjustable slit. By
moving the inner tube back and forth, the instrument may

Fig. 284.

be so focused that parallel rays will fall upon prism A. This
instrument has no telescope. By varying the kind of glass
used in the different prisms,^ as well as their structure, the
deviation of light from a straight path in passing through
them is overcome, while the dispersion is preserved. On
account of the directness of the path of light through it, this
instrument is called a direct-vision spectroscope.
337. Bright line spectrimis.

Experiment 7. — Open the slit about one-sixteentli of an inch wide,
by turning the milled ring M (Fig. 285), and look through the spectro-
scope at the sky (not at the sun, for its light-
waves are too intense for the eye); you will
see the solar spectrum.

Experiment 8. — Repeat the last experiment
with a candle, kerosene, or ordinary gas-flame,

Fig. 285.

and you will obtain similar results.

1 A and C are crown-glass, and B is flint-glass. See footnote, p. 372.

BRIGHT LINE SPECTRUMS. 377

Experiment 9. — Take a piece of platinum wire 2 inches long. Seal
one end by fusion to a short glass tube for a handle. Bend the wire at
a right angle. Dip a portion of the wire into a strong
solution of common salt, and support it by a clamp in a
the midst of the almost invisible and colorless flame of (iliL.^^^^
a Bunsen burner (Fig. 286). Instantly the flame ^11 ^""""^^^
becomes luminous and colored a deep yellow. Examine |
it with a spectroscope, and you will find, instead of a I
continuous spectrum beginning with red, only a bright, M, ^^^^^
narrow line of yellow, in the yellow part of the spectrum, ^^J^^^^l
next the orange. Your spectrum consists essentially of ^^^^''~
a single! bright yellow line on a comparatively dark Fig. 286.

ground (see Sodium, Plate I, frontispiece).

Experiment 10, — Heat the platinum wire until it ceases to color the
flame, then dip it into a solution of chloride of lithium, and repeat the last
experiment. You obtain a carmine-tinted flame, and see through the
spectroscope a bright red line and a faint orange line (see Lithium, Plate I).

Experiment 11. — Use potassium hydrate, and you obtain a violet-
colored flame, and a spectrum consisting of a red line and a violet line
(the latter very difiicult to see even with the best instruments). Use
strontium nitrate, and obtain a crimson flame, and a spectrum consist-
ing of several lines in the red and the orange and a blue line (see Potas-
sium and Strontium, Plate I).

Experiment 12. — Use a mixture of several of the above chemicals,
and you will obtain a spectrum containing all the lines that characterize
the several substances.

Every chemical compound used in the above experiments
contains a different metal, e.g. common salt contains the
metal sodium ; the other substances used successively con-
tain respectively the metals lithium, potassium, and stron-
tium. These metals, when introduced into the flame, are
vaporized, and we get their spectrums when in a gaseous
state. All incandescent gases, unless %inder great 'pressure,
give discontinuous, or bright line, sjpectrunfis, and no two gases
give the same spectrum.

1 Spectroscopes of higher dispersive power show that the sodium line is really a
double line divided by a narrow interval.

" It is not a hypothesis, but a reality, that sodium vapor has two independent
vibrations, whose periods differ by about ^q^otj of each other." —Lord Kelvin.

378 ETHER DYNAMICS.

338. Spectmm analysis.

A vibrating molecule embedded in ether emits waves, the length
of which depends on the rate of vibration, and the waves of different
lengths produce the different color sensations. Now, like a tuning
fork, a free molecule of every substance has its own definite period
or periods of vibration, and accordingly sends out light of a certam
definite color or of a few definite colors, just as the fork emits sound
of a certain definite pitch, with sometimes a few harmonics. For
example, every molecule of sodium or of lithium vibrates in the
same way, and always has vibrated in the same way, whether it
exists in the sun, in the earth, or in a distant star. The same is
true of every other kind of matter ; each has its own rates of vibra-
tion, and hence each produces its own bright line spectrum corre-
sponding with its peculiar rates of vibration. Hence has arisen a
new chemical analysis, wherein substances are detected simply by
observing the rates of vibration of their molecules {i.e. the bright
lines of their spectrums), a branch of physical chemistry called
spectrum analysis.

It is only in the gaseous state, however, that the molecule is free
to exhibit its special rate of vibration ; when they are packed closely
together in a solid or liquid, their motions are cramped, their perio-
dicity is lost, and all manner of vibrations are induced. Hence
spectrums of solids and liquids are continuous, i.e. the rates of
vibrations are so many in number as to leave no gaps in their
spectrums.

Many chemical compounds are decomposed into their elements,
and the elements are rendered gaseous at a temperature that is at,
or below, the temperature necessary for incandescence. In that
case the spectrum given is the combined spectrums of the elements.
A compound gas that does not suffer dissociation at the temperature
of incandescence gives its own spectrum, which is generally totally
different from the spectrum of its elements.

339. Reversed or dark line spectrum.

Experiment 13. — Arrange apparatus in a dark room as in Fig. 287.
N is the flange nozzle of a stereopticon (p. 432) containing only the con-
densing lens ; T and S are two tin plates, in the latter of which a slit is
cut. Allow a beam of calcium light to pass through the slit in S, and
thence through the converging lens L and the prism P, and form a spec-
trum on a screen, H. Hold in the flame of a Bunsen burner, B, a pellet

REVERSED OR DARK LINE SPECTRUM.

379

of sodium ; it burns vividly, and the light has to pass through the in-
tensely yellow flame. We should naturally expect that the yellow of the
spectrum would now be more intensely illuminated, but, instead, a dark
band in the yellow now appears. It is not really black, but compara-
tively dark.

Next hold the plate T between the burner and the condensers so that
the calcium light may be cut off from the upper portion of the slit,
leaving the light from the sodium flame alone to pass through this part
of the slit, The spectrum R formed by this part consists of a bright
yellow line on a dark ground, being the radiation spectrum of sodium.
(It should be borne in mind that the image of the slit is inverted.) The

Fig. 287.

other half, A, shows a dark line on the continuous spectrum. We thus
have, contiguous to each other, the bright line spectrum of sodium and
its reversed^ dark line, or absorption spectrum. If you use salts of lithium,
potassium, strontium, etc., in a similar manner, you will find in every
case your spectrum crossed by dark lines where you would expect to find
bright lines.

It thus appears that the vapors of different substances absorb
or quench the very same rays that they are capable of emitting
when made self-luminous; very much, it would seem, as a
given tuning-fork selects from various sounds only those of

380 ETHER DYNAMICS.

a definite wave-length corresponding to its own vibration-
period. The dark places of the spectrum receive light in
full force from the salted flame.; but the light is so feeble,
in comparison with those places illuminated by the calcium
light, that the former appear dark by contrast. Light trans-
mitted through certain liquids (as sulphate of quinine and
blood) and certain solids (as some colored glasses) produces
band spect-rums. These spectrums are obtained only when
light passes through mediums capable of absorbing rays of
certain wave-length ; hence, they are commonly called absorp-
tion spectrums. Since a given vapor causes dark lines pre-
cisely where it would cause bright lines if it were itself the
only radiator of light, dark line spectrums are frequently
called reversed spectrums. There are then three kinds of
spectrums : continuous spectrums, produced by luminous solids,
liquids, or, as has been found in a few instances, gases under
great pressure ; bright line spectrums, produced by luminous
vapors ; and absorption spectrums, produced by light that has
been sifted by certain mediums.

340. Fraunhofer^s lines. — The spectrum of sunlight is
observed to contain a large number of dark lines transverse
to its length. These were first observed by Wollaston (1802),
and were mapped by Fraunhofer (1814) who distinguished
several of the more prominent ones by letters of the alpha-
bet ; hence the dark lines of the solar spectrum have received
the Dame of Fraunhofer' s lines.

So far as discovered, no two substances have a spectrum
consisting of the same combination of lines ; and, in general,
different substances very rarely possess lines appearing to
be common to both. Hence, when we have once observed and
mapped the spectrum of any substance, we may ever after be
able to recognize the presence of that substance when emit-
ting light, whether it is in our laboratory or in a distant
heavenly body.

fkaunhofer's lines. 381

The spectroscope, therefore, furnishes us a most efficient
means of detecting the presence (or absence) of any elemen-
tary substance, even when it is combined or mixed with other
substances. It is not necessary that the given substance
should exist in large quantities ; for example, the fourteen-
millionth part of a milligram of sodium can be detected by
the spectroscope. Substances which are not easily converted
into vapors at low temperatures may be placed between the
poles of an electric battery or an induction coil. The heat
generated by electricity will vaporize most substances. Thus
the spark passing between two copper electrodes will vaporize
a portion and show the copper lines, between iron electrodes
the iron lines, etc. After maps of the spectrum of all known
substances have been made out, if, on examination of a com-
plex substance, any new lines should at any time appear in
the spectrum, it would indicate the presence of a substance
hitherto undiscovered. It was thus that the elements cae-
sium, rubidium, thallium, and indium were discovered.^

341. General remarks.

Gases or vapors when sufficiently heated to become luminous
emit, under ordinary pressure, color rays which are dispersed into
an interrupted spectrum of bright lines ; with increasing pressure
and density these lines, spread into diffuse luminous bands, and
finally form a continuous spectrum. Gases are rendered luminous
usually by passing electric sparks through glass tubes enclosing
them. Substances whose great volatility interferes by causing
evaporation before the substpvnce attains the temperature of incan-

1 Fraunliof er's lines, designated by letters of the alphabet, beginning at the red
end of the spectrum, have the follo^ving wave-lengths, expressed in million ths of a
millimeter :

Line. Wave-length. Line. Wave-length.

A 762.1 Ej 527.0

B 688.4 b, 518.3

C . 656.3 F 486.1

Di 589.6 G 430.7

Da 589.0 H, 396.8

H, 393.3

382 ETHER DYNAMICS.

descence, such as most metalloids, give no flame emission spectrums.
These are commonly rendered luminous in glass tubes called Plticker
tubes (Fig. 288).

Colored liquids are usually placed in glass cells with flat parallel
sides, or of wedge shape so as to allow the examination of different
thicknesses of the liquid. The cells are placed between —>.
a bright flame and the slit (or better in reflected sunlight), A
and the result is absorption spectrums of the coloring ( j
matter, consisting of dark bands. Thus blood in its
normal or healthy state is readily recognized by the
absorption spectrums of its coloring matters and their
modifications by absorption of gases, i Aniline blue
shows a very dark absorption band from wave-length
656 (C) to 550, gradually becoming lighter from there to
520 (just beyond b').

The spectroscope is very useful to the pathologist in

examining diseased blood, in detecting albumen in urine, — '

Fig 288
in investigating supposed cases of poisoning, etc. ; to the

merchant in distinguishing certain liquids such as wine, beer, etc.

in the normal state and the adulterated state.

342. Solar and stellar chemistry and plujsics.

The spectrum of iron has been mapped to the extent of more
than 600 bright lines. Of these, Kirchhoff succeeded in showing
the coincidence of 460 with dark lines of the solar spectrum. Can
there be any doubt of the existence of iron in the sun ? By exam-
ination of the reversed spectrum of the sun, we are able to deter-
mine with certainty the existence there of sodium, calcium, copper,
zinc, magnesium, hydrogen, and many other known substances.
Again, from our knowledge of the way in which a reversed spectrum
can be produced, we may conclude that the sun consists of a lumi-
nous solid, a liquid, or an intensely heated and greatly condensed
gas (called a p/io^osp/zere), and that this nucleus is surrounded by an
atmosphere of cooler vapor, in which exist at least all the substances

lA solution of fresh blood gives two easily distinguishable dark bands in the
green. But blood is capable of existing in different stages of oxidation which are
distinguishable by difference of color and corresponding difference of spectrum. By
means of a microspectroscope (a combination of microscope and spectroscope) it is
claimed that the thoiisandth part of a grain of blood is easy of detection, and its pres-
ence may be detected in stains that have been kept a very long time ; hence this in-
strument often becomes of great importance in criminal trials.

EFFECT OF MOTION IN THE RADIATING BODY. 383

just named. The moon and planets that are visible only by reflected
sun-light give the same spectrums as the sun, while those that are
self-luminous give spectrums which differ from the solar spectrum.

343. Effect of ^notion in the radiating hody.

If the radiating body be in motion, either to or from the observer,
obviously the effect of this motion will be to shorten the wave-length
in the former case and to lengthen it in the latter (compare §173).
This will tend to move the spectrum lines toward the more re-
frangible end in the first instance and toward the less refrangible
end in the latter. Thus from the displacement of hydrogen lines in
the absorption spectrum Young computes that the greatest speed
upon the sun observed by him is 400 kilometers per second.
Pickering (Harvard Univ.) has observed that the Tc line in the spec-
trum of j8 Aurigae is alternately single and double at intervals of
about seventeen hours, thus showing this star to be double, each
part revolving about the other in less than four days at a speed
of 240 kilometers per second.

The telespedroscope (a combination of telescope and spectroscope)
has disclosed to us a much-coveted knowledge of the true nature,
chemical composition, and physical condition of those points of
light called the fixed stars, immensely more remote and less bright
than the planets. In like manner nebulae, comets, and meteors
have been investigated, and valuable knowledge has been obtained
as to their physical constitution.

344. Distribution of energy in the spectrum. — The energy
of ether waves is capable, as has been before observed, of pro-
ducing calorific, luminous, or chemical effects, according to
the nature of the bodies upon which it falls. When a sensi-
tive thermoscope is passed along the spectrum, heat effects
are observed throughout the visible spectrum, and for con-
siderable distances beyond at each extremity. All ether
waves are capable of producing heating effects.

It thus appears that the solar spectrum is not limited to the
visible spectrum, but extends beyond at each extremity, and
spectroscopic analysis, besides sifting the waves of one color
from those of another, is able to sift out rays which do not

384 ETHER DYNAMICS.

produce the sensation of light from those which do. Those
rays that lie beyond the red are called the infra-red rays, while
those that lie beyond the violet are called the ultra-violet rays.
The infra-red rays are of longer vibration period, and the
ultra-violet of shorter period, than the luminous waves.

Inasmucli as glass largely absorbs the energy of ether waves of
certain lengths, it is customary in studying heat spectrums to use
lenses and prisms of rock saft, since this substance transmits waves
of all lengths with great freedom.

In a prismatic spectrum obtained by the use of a rock salt prism,
the maximum heating effect for the solar spectrum is in the infra-
red. Langley finds, however, that in the normal spectrum the maxi-
mum heating effect in the solar radiation coincides quite closely
with the maximum luminous effect which is in the orange-yellow.

Chemical effects are produced by rays of all refrangibilities.
A photograph can be taken of all portions of the visible spectrum,
and the photographic spectrum may extend far beyond the visible
spectrum in the ultra-violet, and even the infra-red rays may be
photographed. Ordinary silver salts are decomposed by rays ex-
tending from the green upward, while the decomposition of carbon
dioxide is aided chiefly by rays of lower refrangibility.

345. Only one kind of radiation. ■ — The fact that radiant
energy produces three distinct effects, — viz. luminous, heat-
ing, and chemical, — has given rise to a prevalent idea that
there are three distinct kinds of radiation.^ There is, how-
ever, absolutely no proof that these different effects are pro-
duced by different kinds of radiation. Science recognizes in
radiations no distinctions but periods, wave lengths, and wave
forms. The same radiation that 2>^'oduces vision can generate
heat and chemical action. The fact that the infra-red and ultra-
violet rays do not affect the eye does not argue that they are of
a different nature from those that do, but it does show that

1 One great service which the diifraction spectrum (see § 361) has rendered to science
is the aholishment of all these imaginary independent existencies — heat, light, acti-
nism, etc., and the suhstitiition for them of the far simpler conception of vibratory
motions of ether differing only in rate of vibration, the diversity of effects produced
depending on the quality of the surface on which they fall.

PHOSPHORESCENCE. 385

there is a limit to the susceptibility of the eye to receive im-
pressions from radiation. Jnst as there are sound-waves of
too long, and others of too short period to affect the ear, so
there are ethereal waves, some of too long, and others of too
short period to affect the eye.

346. PJiosphorescence. — There is a class of substances such
as the sulphides of calcium, strontium, etc., which after
several hours' exposure to light-waves absorb their energy
(i.e. their molecules acquire sympathetic vibrations) without
becoming hot, and in return emit light-waves, which are quite
perceptible in a dark room for several hours after the ex-
posure. This property of shining in the dark after having
been exposed to light-waves is termed pJiosphorescence. A so-
called Imninous jpahit is prepared and applied to certain parts
of bodies that are exposed to sunshine during the day ; at
night those parts to which the paint is applied are alone
luminous. This paint may be used for a variety of purposes,
such as rendering luminous danger signals, door numbers and
plates, etc.

347. Fluorescence.

There is another class of substances which are acted upon in a
somewhat similar and yet somewhat different manner. The vibra-
tions of the extremely small ether-atoms may set up in the more pon-
derous molecules of matter slower (forced) vibrations. We have an
example of this when light-waves are absorbed by a body and it in
turn emits only the longer, invisible waves. Much as short, choppy
waves acting upon a vessel anchored at sea impart a slower pitching
and rolling of the vessel, and these in turn new and slower waves in
the water, the invisible ultra-violet waves may give rise to dis-
turbances of the interior molecules of a body impinged upon, and
give rise to other waves which are not so frequent as to be invisible.
In this case the body emits light from within, and in some cases
continues to emit light for some seconds after the light is shut off.
This phenomenon is known by the name fluorescence, from fluor-
spar, one of the first substances in which it was observed. Among
fluorescent substances are aesculin (derived from the bark of the

386 ETHER DYNAMICS.

horse-chestnut), quinine sulphate, sulphides of barium and calcium,
diamond, uranium glass, etc. These substances when illuminated
by certain rays of the spectrum in a darkened room shine from
within with a lustre of their own, each showing its own special
color, a color not by any means the same as the natural color of
the body itself as seen in the open white light of day. This is best
accomplished by means of the light of electric discharges. For this
purpose solids are usually enclosed in the so-called Geissler tubes
and an electric current is passed through them.

348. Calorescence.

On the other hand, slow waves may be transformed into more
rapid ones. A beam of light passing through a solution of iodine
in carbon bisulphide loses all its visible rays and only the long
infra-red rays pass through. If these be brought to a focus by
means of a lens upon a piece of platinum, the platinum will become
luminous, and the light emitted therefrom when examined by
means of a prism shows a continuous spectrum. This elevation in
the rank of wave-length is called calorescence.

Section IX.

COLOR.

349. Color hy absoriotion. — Color is a sensation ; it has no
material existence. The term ''yellow light" means, pri-
marily, a particular sensation; secondarily, it means the
physical cause of this sensation, i.e. a train of ether-waves
of a particular frequency. '' All objects are black in the
dark " ; this is equivalent to saying that without light there
is no color.

Experiment 1. — By means of a porte-lumiere introduce a beam of
sunlight into a dark room. With the slit and prism form a solar spec-
trum. Between the slit and prism introduce a deep red glass; all the
colors of the spectrum except the red are much reduced in intensity.

It thus appears that the color of a colored transparent
object, as seen by transmitted light, arises from the unequal
absorption of the different colors of white light incident upon

COLOR BY ABSORPTION.

387

it. 'A red glass absorbs less red light than light of other
colors. The color produced by absorption is rarely very
pure, the particular hue of the transmitted light being due
merely to a predominance of certain colors, and not to the
absence of all others. As the absorbing layer is thicker, the
resulting color is purer but less intense.

Experiment 2. — We have found that common salt introduced into a
Bunsen flame renders it himinous, and that the light when analyzed with
a prism is found to contain only yellow. Expose papers or fabrics of
various colors to this light in a darkened room. No one of them except
yellow exhibits its natural color.

Experiment 3. — Hold a narrow strip of red paper or ribbon ^ in the
red portion of the solar spectrum ; it appears red. Slowly move it
toward the other end of the spectrum ; on leaving the red it becomes
darker, and when it reaches the green it is quite black or colorless, and
remains so as it passes the other colors of the spectrum. Repeat the
experiment, using other colors, and notice that only in light of its own
color does each strip of paper appear of its natural color, while in all
other colors it is dark.

These experiments show that the color of a body seen by
light reflected from it depends both upon the color of the
light incident upon it and upon the nature of the body.

If a piece of colored glass, AB (Fig. 289), be held near a
window so as to receive, obliquely, rays of sunlight, a portion
of the light will be reflected by the anterior surface of the
glass, and, falling ^^ ^^

upon the white ceil-
ing, will illuminate
it with white light.
Another portion of
the light will enter
the glass and be re-
flected from the pos-
terior surface ; this light, having entered the glass and

1 Care must be exercised to select only pure colors.

388 ETHER DYNAMICS.

traveled in it a distance a little greater than twice its thick-
ness, will suffer an unequal absorption of its rays, and after
emerging from the glass will, if the glass be blue, illuminate
a neighboring portion of the ceiling with blue light. This
illustrates the method by which pigments afford color. Thus,
the first surface of a water-color drawing reflects the white
daylight. Most of the light reflected to the eye has, how-
ever, passed through the pigment to the white paper beneath,
and being reflected from this, again passes through the layer
of pigment before reaching the eye. With less transparent
pigments the light may be reflected merely by particles of
pigment beneath the surface. The color of paints and pig-
ments is, therefore, due to the rays which they absorb least
readily. When we paint our houses we do not apply color to
them; we apply substances which have the property of ab-
sorbing or subtracting from white light largely all the colors
except those which we would have our houses appear. This
is technically called selective absorption.

The color of bodies thus depends generally upon their mo-
lecular structure. Different bodies quench different portions
of the complex sunlight. The unqueiiched light determines
the color of a body.

The molecular action in the case of absorption is this. The
molecules of the substance receiving incident light are capable of
vibrating in unison with certain ether waves. The energy of cer-
tain of the ether-waves is employed in setting up molecular vibra-
tions in the substance, thus raising its temperature, while the
energy of waves of different period is propagated through the me-
dium without producing this effect.

350. Opalescence. Sky colors.

Experiment 4. — Dissolve a little white castile soap in a tumbler of
water ; or, better, stir into the water a few drops of an alcoholic solution
of mastic, enough to render the water slightly turbid. Place a black
screen behind the tumbler, and examine the liquid by reflected sunlight,

MIXING COLORS. 389

— the liquid appears to be blue ; examine the liquid by transmitted sun-
shine, — it now appears yellowish red.

Experiment 5. — Pour some of the turbid liquid into a small test-tube,
and examine it and the tumbler of liquid by transmitted light ; the former
appears almost colorless, while the latter is deeply colored.

When a medium holds in suspension fine particles of mat-
ter, the shorter light-waves are most abundantly reflected,
giving a blue color. The blue is purer as the particles are
smaller. Objects seen through such mediums appear of the
complementary hue (see § 354). This phenomenon is called
opalescence. It accounts for the blue of watery milk, opa-
lescent glass, smoke, and the sky.

Skylight is reflected light. The minute particles (of water,
probably) that pervade the atmosphere, like the fine particles
of mastic suspended in the water, reflect blue light; while
beyond the atmosphere is a black background of darkness.
But we must not, from this, conclude that the atmosphere is
blue ; for, unlike blue glass, but like the turbid liquid, it
transmits yellow and red rays freely, so that seen by re-
flected light it is blue, but seen by transmitted light it is
yellowish red.

The remarkable "yellow days" of the summer of 1883 are
explained in this way. The atmosphere on this continent
was very turbid during those days.

When the sun is near the horizon, its rays travel a greater
distance in the air to reach the earth than when it is in the
zenith (see Fig. 263) ; consequently, there is a greater loss by
absorption and reflection in the former case than in the latter.
But the yellow and red rays suffer less destruction, propor-
tionally, than the other colors ; consequently, these colors
predominate in the morning and evening.

351. Mixing colors. — A mixture of all the prismatic colors
in the proportion found in sunlight produces white. Can
white be produced in any other way ?

390

ETHER DYNAMICS.

Experiment 6. — On a black surface, A (Fig. 290), lay two small rec-
tangular pieces of paper, one yellow and the other blue, about two inches
apart. In a vertical position between these papers,
and from 3 inches to 6 inches above them, hold a slip
of plate glass, C. Looking obliquely down through
the glass j^ou may see the blue paper by transmitted
light-waves and the yellow paper by reflection. That
is, you see the object itself in the former case, and
the image of the object in the latter case. By a little
manipulation the image and the object may be made
to overlap each other, when both colors will ap-
parently disappear, and in their place the color which
is the result of the mixture will appear. In this case
it will be white, or rather, gray^ which is white of a
If the color be yellowish, lower the glass ; if

Fig. 290.

low degree of luminosity.
bluish, raise it.

Experiment 7. — With the rotating apparatus, rotate the disk (Fig. 291)
which contains only yellow and blue. The colors {i.e. the sensations) so
blend in the eye as to produce the sensation of gray.

Fig. 292 represents "Newton's disk," which contains the
seven prismatic colors arranged in a jjroper proportion to
prodnce gray when rotated.

Fig. 291.

Fig. 292.

Fig. 293.

In like manner, you may produce white 15y mixing purple
and green ; or, if any color on the circumference of the circle
(see Complementary Colors, Plate I) be mixed with the color
exactly opposite, the resulting color will be white. Again,
the three colors, red, green, and violet, arranged as in Fig.
293, with rather less surface of the green exposed than of

MIXING PIGMENTS. 391

the other colors, will give gray. Green mixed Avith red,
in varying proportions, will produce any of the colors
in a straight line between these two colors in the diagram
(Plate I); green mixed with violet will produce any of the
colors between them ; and violet mixed with red gives
purple.

All colors are represented in the spectrum, except the
purple hues. The latter form the connecting link between
the two ends of the spectrum. Our color chart (Plate I) is
intended to represent the sum total of all the sensations of
color. By means of this chart we may determine the result
of the (optical) mixture of any two colors, as follows : Find
the places occupied upon the chart by the two colors Avhich
are to be mixed, and unite the two points by a straight line.
The color produced by the mixture will invariably be found
at the center of this line.

352. Mixing pigments.

Experiment 8. — Mix a little of the two pigments, chrome yellow and
ultramarine blue, and you obtain a green pigment.

The last three experiments show that mixing certain colors,
and mixing pigments of the same name, may produce very
different results. In the first experiments you mixed colors ;
in the last experiment you did not mix colors, and we must
seek an explanation of the result obtained. If a glass vessel
with parallel sides containing a blue solution of sulphate of
copper be interposed in the path of the light-waves which
form a solar spectrum, it will be found that the red, orange,
and yellow waves are cut out of the spectrum, i.e. the liquid
absorbs these waves. And if a yellow solution of bichromate
of potash or picric acid be interposed, the blue and violet
waves will be absorbed. It is evident that, if both solutions
be interposed, all the colors will be destroyed except the
green, which alone will be transmitted ; thus : — -

392

ETHER DYNAMICS.

Cancelled by the blue solution, ^ ^ G B V.

Cancelled by the yellow solution, K O Y G |^ y.

Cancelled by both solutions, ]^ ^ G ^ /.

In a similar manner, when white light strikes a mixture of
yellow and blue pigments on the palette, it penetrates to some
depth into the mixture ; and, during its passage in and out,
all the colors except the green are destroyed ; so the mixed
pigments necessarily appear green. But when a mixture of
yellow and blue waves enters the eye, we get, as the result
of the combined sensations produced by the two colors, the
sensation of white ; hence a mixture of yellow and blue gives
white.

The color square 3 (Plate I) represents the result of the
mixture of pigments 1 and 2 ; while 4 represents the result
of the optical mixture of the same colors.

353. Theories of color vision.

Brewster (1831) presented the hypothesis that all colors are formed
by the union of three primaries, — red, yellow, and blue, — which
together compose white light, and give, by combinations in twos,
the hues, orange, green, purple, etc. This hypothesis was based on
the mixing of pigments. But the actual addition of colors does not
give the same result as the mixing of pigments, as has been shown.

The generally accepted theory of color-vision is that of Dr. Young
(1801-2), verified by Maxwell and Helmholtz. It supposes the ex-
istence of three color sensations, red, green, and violet. These ex-
cited simultaneously,
and with proper in-
tensities, produce the
sensation of white
light. Combined in
twos, they produce
the remaining color
sensations. Thus red
and green sensations
combined give yellow
or orange ; green and violet give blue, etc. The longer light-waves
excite the sensation of red ; together with those somewhat shorter,

COMPLEMENTARY COLORS. 393

they excite both red and green, thus giving yellow, and so on.
Strictly speaking, light-waves of any length excite all three sensa-
tions ; but usually either one or two of them greatly predominate.

The relative intensities of the various color-sensations throughout
the spectrum as obtained from actual measurement by Maxwell are
shown in Fig. 294.

354. ComjjleTnentary colors.

Experiment 9. — On a piece of gray paper lay a circular piece of blue
paper 15 mm in diameter. Attach one end of a piece of thread to the
colored paper, and hold the other end in the hand. Place the eyes with-
in about 15 cm of the colored paper, and look steadily at the center
of the paper for about fifteen seconds ; then, without moving the eyes,
suddenly pull the colored paper away, and instantly there will appear on
the gray paper an image of the colored paper, but the image will appear
to be yellow. This is usually called an after-image. If yellow paper be
used, the color of the after-image will be blue ; and if any other color
given in the diagram (Plate I), the color of its after-image will be the
color that stands opposite to it.

This plienomenon is explained as follows : When we look
steadily at blue for a time, the eyes become fatigued by this
color, and less susceptible to its influence, while they are fully
susceptible to the influence of other colors ; so that when they
are suddenly brought to look at white, which may be regarded
as a compound of yellow and blue, they receive a vivid im-
pression from the former, and a feeble impression from the
latter ; hence the predominant sensation is yellow. Any two
colors which together produce white are said to be comple-
mentary to each other. The opposite colors in the diagram
(Plate I) are complementary to one another.

The complement of green is purple, — a compound color
not existing in the spectrum.

The eye gives no direct knowledge that the composition of
light produces the sensation of white. Whether this sensa-
tion is produced by the coexistence of all the rays of the
visible spectrum, by a combination of light of two comple-

394 ETHER DYNAMICS.

mentary colors, or by a mixture of rays of tlie three primary
colors, can be determined only by some process of physical
analysis.

355. Effect of contrast. — When different colors are seen at
the same time, their appearance differs more or less from that
observed when they are seen separately. Thus a red object
(e.g. a red rose) appears more brilliant if a green object be
seen in juxtaposition with it. Such effects are said to be due
to contrast.

When any two colors given in the circle (Plate I) are
brought in contrast, as when they are placed next each
other, the effect is to move them farther apart in the color
scale. For example, if red and orange be brought in con-
trast, the orange assumes more of a yellowish hue, and the
red more of a purplish hue. Colors that are already as far
apart as possible, e.g. yellow and blue, do not change their
hue, l^ut merely cause each other to appear more brilliant.

356. Color-blindness. — In this defect in vision, one of the
three color sensations is either wanting or deficient, usually
that of red ; so that the colors perceived are reduced to those
furnished by the remaining two sensations, viz., green and
violet. This causes the red-blind person to confound reds,
greens, and grays. In some rare cases the sensation of green
or violet is the one deficient.

Section X.

INTERFERENCE AND DIFFRACTION.

We have already studied wave interference in the case of
sound (see § 178), and must recall that two sets of sound-
waves may neutralize each other and produce silence. As
an example, we cite the phenomenon of beats, in which the
alternate increase and diminution of intensity is due to the

young's theory. 395

interference of two sets of sound-waves in the same and oppo-
site phases respectively. If radiation be a wave motion,
similar phenomena ought to occur under proper conditions.

357. Yoiong^s theory. — The earliest authentic experiments,
on the interference of light-waves were made by Dr. Young
in 1801. He admitted a beam of sunlight into a dark
chamber through a very narrow aperture, and in its path
placed a screen having two very small openings quite near
together. When the two overlapping pencils from these
openings were made to fall on a white screen there appeared
a series of bands alternately bright and dark, which disap-
peared when one of the holes was covered.

Although Young ascribed this phenomenon to interference,
and explained it very satisfactorily, with the wave theory as
a basis, his views were by no means universally accepted.

Grimaldi, nearly a century and a half before, had noticed light
and dark fringes at the edges of shadows of small opaque bodies
placed in the path of sunlight admitted through a small hole. This
action was called diffraction (see § 360), and was afterwards studied
by Newton. In view of the emission theory, the phenomenon was
explained by assuming that the light particles experience an at-
tractive or repellent force as they come near the edges of bodies.
Believers in the latter theory then contended that Young's experi-
ment was simply one of diffraction.

To remove these objections it was necessary to devise means of
producing the same result, but entirely independently of apertures
and opaque bodies. This was first done by Fresnel,! who contrived
two most ingenious experiments for producing the light and dark
bands, m which the results could be accounted for only by assuming
Young's theory of interference to be correct.

Let us now consider what was observed by Young. A and B
(Fig. 295) are two illuminated apertures very near together, and
emitting waves in all directions. When these two sets of waves
arrive at any point P on the screen, they will be in the same phase
only if the distance of P from one of the sources, say B, is one or

1 A full account of Fresnel's experiments is given in " The Theory of Light," by
Preston, §§ 86, 87

396

ETHER DYNAMICS.

Fig. 295.

more complete wave-lengths more than the distance P A. In this
event the two waves will conspire to increase the illumination at P,
and this point will be on a bright band.

On the other hand, if P B - P A be equal
to a half wave-length or any odd number of
half wave-lengths, the waves from A and B
will arrive at P in exactly opposite phases
and destroy one another, and P will be a
point on a dark band. At intermediate
positions for P, P B - P A will not equal
any whole number of half wave-lengths,
and hence the waves will meet in neither
the same nor opposite phases, and the il-
lumination at P will be intermediate in its
intensity between that of the brightest and
darkest points. Thus the bands will shade off imperceptibly into
one another.

In the above discussion we have assumed the light to be mono-
chromatic, i.e. all of the same wave-length. Let O M be a perpen-
dicular to the screens half way between A and B ; then the system

of bands evidently is
arranged symmetrical-
ly about M, this point
being on a bright band.
It is easy to see that
the distance of any
given band from M
bears a simple relation
to the wave-length,
and hence the bands will be closer together for short waves than for
long ones, as shown in Fig. 296.

If composite light be used, we should expect the bands to be colored,
which is the case for a few of them near the central band, which is
white ; the edges nearer M are blue, while the outer edges are red.
As each color gives rise to a separate system of bands, the red
ones being broadest and the violet ones narrowest, it will happen
that after a few alternations a red band and a violet one will fall at
the same place ; or the dark spaces of one system will be filled by
the bright parts of another. Soon, then, as we recede from M, the
bands become less distinctly marked, and finally merge into one
another and fade into uniform illumination.

■awm nil

■IIIIIIH

Fig. 296.

COLORS OF THIN PLATES. 397

358. Colors of thin x^lates. — Everybody is familiar with
the beautiful color effects produced when ordinary white light
falls upon a thin film of a transparent substance, such as a
soap bubble or a film of oil on water. This is a case of inter-
ference. Some of the light comes to the eye reflected from
the front or nearer surface of the film ; another portion has
entered the medium and been reflected from the back surface.
When it emerges, it has been retarded an amount depending
on the distance it has traveled in the film. We might expect
that if the retardation were an even number of half wave-
lengths, the two portions of light would be in a condition to
interfere in the same phase, and that the effect would be
increased ; while if the retardation were an odd multiple of
half wave-lengths, the interference would be destructive, and
darkness would result.

The fact is exactly the reverse of this, since by the act of
reflection in the denser medium the phase of the wave is
reversed and the result is as if the wave had been retarded
another half wave-length.

If the eye view such a film in monochromatic light, the
portion entering the film will be retarded varying amounts
depending on the angle of incidence and on the thickness.
The film will, therefore, be crossed by bright and dark
bands.

If, however, composite light be used, cooperative or de-
structive interference evidently cannot take place for the
different colors at the same points, and the familiar iris-
colored bands result.

Experiment 1. — Press firmly together two poHshed pieces of thick plate
glass. A number of colors will be seen arranged in a certain order, and
forming curves more or less regular around the point of pressure. Ex-
plain.

359. ]\^ewto7i's rings.

Newton's method of studying these colors was very simple
and effective, and the phenomena exhibited are known as ' ' New-

398 ETHER DYNAMICS.

ton's rings," If a convex lens of very small curvature be placed
firmly upon a piece of plate glass (Fig. 297) the film of air between
the two increases in thickness from the center radially, and hence

Fig. 297. Fig. 298.

beautiful circular interference bands are shown, having the point
of contact as their center (Fig. 298).

360. Dijfraction.

This phenomenon, first observed by Grimaldi, and already referred
to (see §357), occurs when light passes through a very narrow aper-
ture or close to the §dge of an opaque body.

Newton's strong reason for rejecting the wave theory was that
light, as he supposed, did not go round corners as sound does.
Closer examination, however, shows that the cases are similar if
consistent conditions be maintained. Light really does bend round
corners, while on the other hand, well defined sound shadows may
be cast by sounds of sufficiently short wave-length. In other words
sound bends round corners very much more readily than light,
merely because its wave-length is so much greater in comparison
with the size of the obstacle.

The true explanation of diffraction phenomena was given by
Fresnel, who attributed them to the mutual interference of the
secondary wavelets which diverge from the primary or main wave-
front as it meets the obstacle or edges of the aperture ; just as the
cases of interference previously described are due to the mutual in-
terference of two waves. For a fuller description of the cases of
diffraction and their treatment, the student is referred to special
works on optics.

361. The Diffraction Grating}

When a distant source of light is viewed through a system of very
narrow rectangular openings, a central image is seen, and on either
side of it there are several highly colored spectral images, increasing

iFor a more complete treatment of gratings, see Theory of Light, by Preston.

THE DIFFRACTION GRATING.

399

in breadth but diminishing in brightness as they recede from the
center. A device like the above is known as a diffraction grating,
and may be produced by ruling with a diamond fine lines on a
piece of glass. ^ Tlie lines form the opaque portions of the grating,
while the spaces between them are the slits through which the light
passes. The effect is most marked when the opaque parts and
transparent parts are equal.

A magnified section of a grating perpendicular to the Imes is
shown in Fig. 299. Let a h represent a line and adjacent slit in
such a position with
reference to the point
P, where the eye is
placed, that P 6-P a
equals one wave-
length. It is clear
that the portion of the
incident wave (sup-
posed to be homoge-
neous) corresponding
to a & may be divided
into two nearly equal
parts which would de-
structively interfere at
P if the grating were

not present. The effect of the grating then is to intercept one part
of these interfering portions and enable the other to become visible.
A bright band will then be seen in the direction P h.

The same will happen for all similar parts of the grating provided
the distances of their extremities from P differ by a whole number
of wave-lengths. Thus there will appear a succession of bright
bands at increasing angular distances from P M. It should be
noticed in studying this figure that a 6 is extremely small compared
with P M, and therefore the figure is necessarily distorted for
■convenience.

If the light incident on the grating be composite, evidently the
angle a, indicating the direction from P M of the bright band, will
be less for the short waves than for the long ones. Therefore in

Fig. 299.

iMany of the fine gratings of Eowland have 20,000 lines to the inch. These, how-
ever, are now usually ruled on speculum metal, as glass is apt to injure the diamond
point.

400

ETHER DYNAMICS.

this case the resulting band will be colored — the inner edge blue,
the outer one red, the portion lying between, yellow and green.

Such a phenomenon is called a diffraction spectrum. It is what
is known as a normal spectrum^ because it exhibits the colors, or
Fraunhofer lines which locate them, always in their true order and
separated by spaces bearing a simple relation to their wave-lengths.

This is not the case in spectrums produced by refraction. The
rays at the red end are crowded together and condensed out of all

B c

BC D E

G H

Fig. 300.

proportion to those at the violet end ; i.e. a given difference in wave-
length causes a much greater separation in the more refrangible parts.
This is called irrationality of dispersion. It is exhibited variously by
different substances. Fig. 300 shows the positions of the principal
Fraunhofer lines as given with prisms (1) of flint glass, and (2) of
crown glass, having the same refracting angle. The difference

A a 3 C

E b F

inH^-

8|0 7|5 7|0 6|5 6|0 5\o 5|o 4J5 f
ajO 7|5 7|0 6|5 6^ ^5. sjo ^5 4|o

D
Fig. 301.

E b

G 7i H^H

between a normal or diffraction spectrum and a prismatic one may
be understood by a glance at Fig. 301. The scales between the
spectrums indicate the wave-lengths in hundred thousandths of a
millimeter.

The grating furnishes a simple means of measuring wave-length.

REFLECTION GRATINGS. 401

Referring again to Fig. 299, in which P & is the direction of the

first bright band, we see tliat the triangles h ak and 6PM are

very nearly similar {a k being a very short arc practically per-

b k & M
pendicular to P 6). Then —r = — — = sin a ; or bk = ab sin a.
' ab bF

But bk is one wave-length, X, and a 6 is - where n is the number

of lines per unit of length on the grating ;" hence X = - sin a.

Experiment 2. — Introduce a borax bead into the flame of a Bunsen
burner, and place them near the wall of a darkened room. Prom a
distance of six or eight meters, view the flame through a grating
(one having from one to two thousand lines to the centimeter pre-
ferred), holding it perpendicular to the line of sight and with the
rulings vertical. Note the position on the wall of the first image of
the flame, and measure in centimeters its distance from the flame.
This divided by the distance from the point of observation ta the
image will be sin a. (In practice it is found to be sufficiently
accurate to measure from the point of observation directly to the
burner. ) If we divide sin a by the number of lines per centimeter
on the grating, the result will be the wave-length in centimeters
for sodium light.

362. Reflection gratings.

Spectrums similar to those already described may be produced
by reflection from a polished surface (usually of speculum metal)
finely ruled with paral-
lel grooves. 1

The beautiful colors
exhibited by the pol-
ished surface of mother-
of-pearl, by the feathers
of certain birds, and
other striated surfaces, /''T~N
are due to wave inter- (^ 1
ference. This was V.L---'
demonstrated in a strik- (^'rating

r. o- T^ •/. Fig. 302.

mg way by Sir David

Brewster, by taking an impression of the surface in wax, when the
indented wax showed all the colors of the original surface.

1 Fig. 302 represents a sectional view of a grating spectroscope.

402

ETHER DYNAMICS.

Section XL

DOUBLE REFRACTION AND POLARIZATION OF LIGHT.

363. Double refraction. — W6 have hitherto assumed that a
ray of light incident upon a transparent body is refracted
according to the law^ of sines. This is the case when the
transparent body is isotropic, i.e. having the same properties
in all directions. There are numerous transparent substances
which fulfil this condition at least approximately, such as
fluids, well annealed glass, etc. On the other hand, there are
numerous transparent substances for which the law of sines
does not hold. When a ray of light enters such a body it is
spli^i into two rays, and this law does not hold for both these
resulting rays ; sometimes it does not hold for either ray.
This gives rise to a group of phenomena known by the term
double refraction, and the substances which affect light in this
manner are called doubly refracting substances. These include
various crystals, animal substances
such as horn and shells, vegetable
substances such as resins and gums,
and certain artificial substances such
as jellies and un annealed glass.

Experiment 1. — Through a card make
a pin-hole, and hold the card so that you
can see skylight through the hole. Now
bring a crystal of Iceland spar i (Fig. 30.3)
between the eye and the card, and look at
the hole through two parallel surfaces of
the crystal. There will appear to be two
holes, with light shining through each. Cause the crystal to rotate in a
plane parallel with the card, and one of the holes will appear to remain
nearly at rest, while the other revolves around the first.

1 This crystal, sometimes called caicite or calc-spar, is found most abundantly in
Iceland. It exhibits the property of double refraction very clearly, and by means of
it this property was first discovered. By cleavage it can always be brought into the
form represented by the diagram, which is called a rhomh.

Fig. 303.

DOUBLE REFRACTION.

403

Fig. 301.

A ray of light, PQ, immediately on entering the crystal is divided into
two parts, one of which, QO, obeys the regular law of refraction ; the
other, QE, does not. The former is called the ordinary ray ; the latter,
the extraordinary ray. In all crystals which
produce this phenomenon there is one direction,
and in some there are two, in which, if an object
be looked at through the crystal, it does not ap-
pear double. If all the edges of a crystal of
Iceland spar (Fig. 304) be equal, and the line con-
necting the two opposite obtuse solid angles, AB,
be cut near each extremity by a plane perpendicular to it, objects viewed
in this line, or in any line parallel with it, do not appear double.

In every direction in which, one looks through, the crystal,
except that parallel to AB, objects seen through it appear-
double (see Fig. 305). The line AB is called the oj^tiGaxis of
the crystal, and is a line around which the molecules of the
crystal appear to be arranged symmetrically. A crystal is
called uniaxial when it has only one optic axis, and biaxial

PATH JlslanJe^

Fig. 305.

when it has two such axes. A plane parallel to this axis and
perpendicular to one of the rhombic faces of the crystal is
called a principal section. The two rays travel with unequal
speeds in the crystal in all directions except in the direction
of the optic axis of the crystal.

The property of double refraction may be imparted per-
manently or transiently to certain substances which do not
naturally possess it. Glass may be given this power by
heating different parts unequally, and also by compression
and bending.

404 ETHER DYNAMICS.

364. Nicol prism.

For certain purposes, such as are indicated in § 371, it is best to
allow only one of the two rays to leave the prism (that, namely, in
the direction of the incident light), and to elimi-
nate the other. The Nicol prism consists of a
crystal of Iceland spar divided diagonally, as
a b (Fig. 306), the two surfaces being cemented
together with Canada balsam. All the faces
of the prism are painted black except the two
end faces. The extraordinary ray, falling
upon the transparent balsam at an angle less
than the critical angle, passes through it, but
the more refracted (or ordinary) ray meets the
balsam at an angle greater than the critical
angle,! and is therefore totally reflected,
\j thrown to one side of the prism, and absorbed

Fig. 306. hy the black paint.

365. Polarizatio7i of light hy double refraction. — On exami-
nation of the two rays resulting from splitting a single ray
by double refraction, it is found that each is unlike a ray of
common light, that each has properties with respect to a fixed
direction, and that these fixed directions for the two rays are
at right angles to each oth^er. In short, a beam of light thus
treated is not alike upon all sides, but has certain relations to
surrounding space other than direction. This property can
be given to light in various ways. To this phenomenon, how-
ever produced, has been given the name polarization.'^

Slices of crystal of the mineral tourmaline, cut in planes
parallel with their axes, are prepared and sold for optical
experiments. If two of these slices similarly situated, as in
Fig. 307, be placed between the eye and a card pierced by a
hole, the hole will be plainly visible. But if one of the slices be

1 The refractive index of Canada balsam is intermediate between tbe indices of the
crystal.

2 Newton came to the conclusion that each of the two rays had two sides ; and
fiom the analogy of this two-sidedness with the two-endedness of a magnet the term
polarization arose.

POLARIZATION OF LIGHT.

405

Fig. 307.

Fig. 308.

slowly rotated in a plane at right angles with the beam of
light, the hole will grow dimmer until the slice has passed
through a quarter of a revolution (as represented in Fig. 308),
when it disappears.
If the rotation be
continued, the hole
reappears, faint at
first, but reaching its
maximum brightness
at the end of another quarter-revolution. Thus, at suc-
cessive quarter-revolutions it is alternately extinguished and
restored.

It appears, then, that light which has passed through one
transparent slice of tourmaline differs so much from common
light, that a second similar slice may act like an opaque body,

and stop it altogether.
The action of the tour-
maline may be com-
^^ib^l^s-g::^! pared to that of a gratin g
(A, Fig. 309) formed of
parallel vertical rods,
which will allow all vertical planes (as a «') to pass, but stops
the planes (as c c') that are at right angles to these rods.
Any plane that has succeeded in passing one grating will
readily pass a second similarly placed. But if the second
grating, B, be turned so that its rods are at right angles to
the first, the plane that has succeeded in passing through the
first grating will be stopped by the second. Light in this
condition is iMlarized ; polarization is either the act of pro-
ducing the change in the light, or the result of the change,
and the instrument used is a polarizer.

In order to understand this phenomenon, it is necessary
to know more of the undulatory theory of light. This theory
supposes that the undulations in ether which constitute light

Fig. 309.

406

ETHER DYNAMICS.

are much like undulations in a cord when one end is shaken
by a hand ^ as seen in Fig. 310. If the hand move vertically,

all the undulations
will lie in a vertical

Fig. 310.

plane ; if the move-
ments of the hand be
horizontal or oblique,
the undulations lie in corresponding planes. So we can
produce these waves on the rope in any plane passing
through the rope, and can change rapidly from one plane to
another. These waves appear differently when viewed from
different sides. It is believed that if we could look endwise
at a ray of light for an instant, we should see the ether
vibrations, as in the figure of the rope, in one plane ; but in
only a thousandth of a second so many million waves reach the

Fig. 311.

eye that there is time for the vibrating particles, which, like
the hand, start the waves, to vibrate in many planes. In an
ordinary beam of light, as it reaches the eye, there are there-
fore undulations in all possible planes, as is partially repre-
sented by the cross section A (Fig. 311). But rectilinear
motion may be considered as the resultant of two similar
motions at right angles to each other. So here, for many
practical purposes, the vibrations may be regarded as taking
place in only two sets of planes at right angles to each other,

iThe vibratory motion wliicli constitutes light must be transverse to the direction
of the ray ; it cannot be in the direction of the ray, for then there would be no dif-
ference between the different sides of the ray ; and the phenomena of polarization
would be unexplainable on this hypothesis.

PLANE OF POLARIZATION. 407

as represented by B of the same figure. Now, when a ray of
light consisting, according to supposition, of undulations in
planes at right angles to one another strikes a slice of tour-
maline, its molecular structure resolves the motion into two
motions, one parallel to, and the other perpendicular to, its
axis. The former of these is transmitted and the other
is absorbed. By this means the undulations are reduced to
those in parallel planes only, as represented in C. The un-
aided eye cannot usually detect any difference between com-
mon and polarized light. An instrument Avhich will enable
the eye to detect polarized light is called an analyzer ; thus
the first slice of tourmaline serves as a polarizer, and the
second slice as an analyzer. A complete polarizing apparatus,
called a polarlscope^ used for observing the phenomena of
polarized light, consists essentially of a polarizer and an
analyzer.

366. Plane of 'polarization.

There are several ways in which it is possible to restrict the vibra-
tions of a ray of light to one plane. The ray in such a case is said
to be '£ilane 'polarized.

It will be shown presently that light is partially polarized by
ordinary reflection. The plane of incidence in which this occurs is
called the pZcme of 'polarization. It is still an open question
whether the vibrations are parallel to or perpendicular to this
plane, the evidence, however, being decidedly in favor of the latter
view.

With a uniaxial doubly-refracting -crystal {e.g. Iceland spar), the
ordinary ray is polarized in a plane containing the incident ray and
the optic axis.

367. Polariscope consisting of two Nicol prisms.

When a ray of light becomes split by double refraction, each of
the resulting rays is found to be plane polarized, one being polarized
in the plane of incidence, the other at right angles thereto.

A and P (Fig. 312) represent two corks having Nicol prisms ex-
tending through them lengthwise, P serving as a polarizer, and A

408

ETHER DYNAMICS.

as an analyzer. If the analyzer A be turned so that its principal
section (see § 363) of d' E is pEirallel to the principal section c d R of
the polarizer P, the ray R which enters P as unpolarized light be-
comes polarized and the extraordinary ray emerging from this
prism will pass freely through the analyzer A. The same happens

if A be turned through an angle of 180° so as to bring the same
planes parallel again. But if A be adjusted so that its principal
section is at right angles to that of the polarizer (as in the lower part
of Fig. 312), then the ray is quenched by the prism A. No light
leaves the analyzer, and accordingly, to an eye placed at the end of
A, the field of vision is dark. The same happens if A be turned
through an angle of 180°. In all cases where the principal sections
are neither parallel nor at right angles, the polarized light entering
the analyzer is separated into ordinary and extraordinary rays, and
the light which emerges from the analyzer varies in intensity with
the angle at which the principal sections of the two prisms are in-
clined to each other.

368. Polarization by reflection.

Malus discovered (1808), while looking through a double-image
prism at the light reflected from a window in the Luxembourg
palace in Paris, that light may be polarized by reflection.

It was subsequently found that the amount of polarization de-
pends on the incident angle. The angle at which the polarization
is a maximum is called the angle of polarization by reflection, which
in the case of glass is between 55° and 56°. Then, of course, if a

POLARIZATION BY REFRACTION.

409

Nicol prism be held in certain positions the reflected light will pass
through it, but at a distance of 90° it refuses to pass. Again, light
which has been polarized by reflection from one glass surface, A
(Fig. 313), will be reflected from another glass surface, B, placed so

Fig. 313.

that the plane of incidence of the polarized ray is parallel to the
plane of polarization, but utterly refuses to be reflected when the
plane of incidence is at right angles to the plane of polarization,
as in B'. So that in polariscopes a plate (better a bundle of plates)
of glass may be substituted for either the polarizer or the analyzer.
Light reflected obliquely from non-metallic smooth surfaces, such
as water, polished furniture, oil paintings, etc., is found on examina-
tion to be partially polarized. Sky-polarization is due to plane-
polarization effected by reflection from very small particles of water
in the atmosphere.

369. Polarization hy refraction.

Not only the reflected portion of the light is polarized, but also
the part that enters the medium and is transmitted. For example,
if A (Fig. 313) be a transparent glass plate, a part of the incident
light will of course pass through, and this part will be found to be
partially polarized, but in a plane perpendicular to the plane of in-
cidence. Further, Arago found that the reflected beam and the
transmitted beam contain the same amount of polarized light. If
the transmitted beam be examined by making it incident upon a
second transparent surface, it will be reflected only if the plane of
" incidence is made parallel to the plane of polarization of the light.

410 ETHER DYNAMICS.

370. Circular and elliptical polarization.

In all polarization thus far treated, we have assumed the vibra-
tions to be in straight lines and all confined to one plane. It can
be shown, 'however, that if two such waves, one a quarter of a
period in advance of the other, be compounded, with their planes
perpendicular to each other, the result will be a wave in which
the ether particles move in circles or ellipses, according as the
amplitudes of the components are equal or not. Such a wave is
circularly or elliptically polarized, and since (as in all wave motion)
the particles move successively, the wave has the form of a helix.
For fuller discussion of this form of polarization and the methods of
producing and detecting it, the student is referred to special works
on light.

371. Chromatic phenomena}

Take the simplest case, in which the polarizer and analyzer are
sections of tourmaline cut parallel to the axis. (A little considera-
tion will show, however, that the explanation holds good for any
other form of analyzer and iDolarizer.) P (Fig. 314) represents the
polarizer and A the analyzer. P and A are supposed to be crossed.
The arrow shows the direction of the ray. The symbols at L, B,
C, D, are intended to indicate the condition of the light in those
positions. L indicates the two plane component waves of which
ordinary light may be considered as composed. One of these is the
plane of the paper, the other a plane at right angles to it, and con-
taining the ray. The action of the polarizer P is to remove one of
these components, producing at B a plane-polarized wave whose
plane coincides with the plane of the paper. This wave, lying in a
plane at right angles to those waves which the analyzer, when in
the position supposed, will allow to pass, is cut off, and a dark field

results. Any wave not parallel
to the plane of B would evi-
dently pass through A, either
== wholly or in part. Suppose
now any doubly-refracting sub-
S ^ stance, as a crystalline film S,

Pj(j 314 to be placed between P and A.

The plane-polarized ray B, ex-
cept in certain special cases, is doubly refracted and separated into

iln tlie explanation of these phenomena we have adopted largely the language of
Prof. Cross of the Mass. Inst, of Technology.

CHROMATIC PHENOMENA. 411

two component plane-polarized rays, indicated, by C, with their
planes of vibration at right angles to each other, and inclined to the
original plane of vibration of B. These rays at C, in which the
vibrations are not parallel to B, and hence not at right angles to the
axis of the tourmaline section A, are not wholly cut off, but, as A
is a doubly-refracting crystal, are again each separated into two
sets of rays ; one set of these having its plane of vibration at right
angles to B and hence parallel to the axis of A, is allowed to pass
as indicated by D, while the other set is cut off. This explains the
illumination of the field.

' ' The light transmitted under these circumstances is generally
colored, for the following reason : In traversing the doubly-refract-
ing crystal S, the components into which B is separated travel with
different velocities. Hence one of these components gains on the
other by an amount which, other things being equal, depends on
the thickness of S. Suppose the gain to be such that on emerging
from S one of the components of C is in advance of the other by
one wave-length of red light, as shown at 0' (Fig. 315). Assuming
a tourmaline section or equivalent analyzer to be used, one compo-
nent only of each pair into which and 0' are resolved passes through
A, and these components (e e\ Eig. 315) are in opposite phases, and
hence interfere. The components o o' are not transmitted by the
analyzer. The red rays are therefore struck out from the white
light of the original beam, and the field appears of a greenish color.
Evidently, if the thickness of S (Fig. 314) be such that one system
gains upon the other an amount
equal to any whole number of wave-
lengths or even number of half
wave-lengths of light of any parti-
cular color, that color will disappear
from the transmitted rays at D. If
the gain be equal to any odd number
of half wave-lengths, the compo-
nents emerging from the analyzer
will have the same phase, and that

color will be present in the trans- ^

.^^ , , 1 Fig. 315.

mitted beam.i

"If a crystal of Iceland spar be substituted for the tourmaline

analyzer A, both systems of rays, D (Fig. 314) and the set at right

1 Read in connection with the last two statements Glazebrook's Physical Optics,
p. 374 ; also Lloyd's Undulatory Theory of Light, pp. 219, 220.

Jk^

412

ETHER DYNAMICS.

angles to D, will be transmitted. In this case the colors of the two
sets of transmitted rays are always complementary ; that is, if one
be red, the other is green ; if one be blue, the other is yellow, etc.
This is evident from an examination of the component plane-waves
issuing from the doubly-refracting crystal of Iceland spar at A
(Pig. 315). It will be seen that when those of one set meet crest
to trough, those of the other set will meet crest to crest. Hence
any color struck out in one set will predominate in the other ; that
is, the colors of the two sets must be complementary. Rotating the
analyzer through 90° changes each color to the complementary
hue."

Fig. 316.

372. Description of a simple polariscope.

T> (Fig. 316) is a plate of glass about 15 cm square, used as a

polarizer. A is the ana-
lyzer, — preferably a Mcol
prism, — so placed as to
view the center of the glass
at the proper polarizing
angle (about 55°). The
prism, mounted in a cork,
should be free to rotate in
its support. S is a piece of
ground glass used to cut off
the images of outside ob-
jects. G is a glass shelf, on which objects to be examined are
placed. The instrument, covered with a black cloth, is placed on a
table with S toward a window.
Experiment 2. — Place on the support G a thin film of selenite or mica,
and slowly rotate the analyzer. A beautiful display of colors will appear.
At a certain point they will appear of maximum brilliancy, then they
will gradually fade away and change into their complementaries. (See
§371.)

373. JUng-sijstem of plates perpendicular to axis.

A different class of appearances is presented when a plate of any
uniaxial crystal is examined in a convergent or divergent pencil of
plane-polarized light. Since the rays are oblique, the thickness of
the plate traversed increases with the angle of -incidence, being least
for a ray parallel to the axis and normal to the plate. In this case

RING-SYSTEM OF PLATES.

413

the ray passes through the plate parallel to the optic axis, and is
not doubly refracted. Let E be the position of the eye immediately
behind the analyzer (not shown
in Fig. 317), S S' the plate, and
OE the direction of the axis.
Now an oblique ray that reaches
the eye from any point P has
been doubly refracted in passing
through the plate, Pa and P 6
representing the component mo-
tions. As the speeds of the
component rays are not equal,
a retardation which is greater
as P is farther from will occur
in- the case of one of these.

In general, the analyzer again
subdivides the rays P a and P b
into two mutually perpendicu-
lar sets, one component only of each set being transmitted. These
latter interfere more or less destructively according to the retarda-
tion that has been suffered by one of them, the interference being
complete for any color when the difference of phase is half a wave-
length for this color. It is clear, then, that the field will be
occupied by a series of concentric rings spectral colored from
the center outward, the appearance being similar to that of
Newton's rings.

Let us now consider two planes through E, one parallel to the
plane of primitive polarization and the other perpendicular to it.
Let their traces on S S' be M M' and N N'. Now all rays travers-
ing the plate in these planes will be unaffected, since the principal
section of the crystal in the one case coincides with the plane of
polarization of the incident signs, and in the other case is perpen-
dicular to it.

* These rays, therefore, will or will not be transmitted by the
analyzer according as its principal section is parallel to or perpen-
dicular to the primitive plane of polarization. In the former case
we shall have a white cross over the ring system, in the latter a
dark one.

A and B (Fig. 318) illustrate these phenomena only in a very
imperfect way, since the coloring of the figures is necessarily
absent.

414

ETHER DYNAMICS.

374. Rotatory polarization.

It was observed by Arago in 1811 that if plane polarized light be
transmitted by a plate of quartz cut perpendicular to the axis, the

plane of polarization is rotated
through a certain angle. The
investigations of Biot showed
this angle to be proportional to
the thickness of the plate and
approximately to the inverse
square of the wave-length.
For example, if such a plate of
quartz be placed between two
crossed nicols, the dark field
immediately becomes bright,
and also colored, if the light
employed be not homogeneous.
That is, the plate has turned
the plane of polarization of the
light, so that when it is inci-
dent upon the analyzer there
is a component of the vibration
which is parallel to the princi-
pal section, and so can be trans-
mitted. As the different colors
are rotated through different
angles, the transmitted compo-
nents are not mixed in the
same proportions as in the in-
cident light, and so the field is
colored, the particular color
Fig. 318. depending on the position of

the analyzer. It was found that some specimens of quartz cause
rotation to the right {i.e. clockwise), and so are called dextrogyrate,
while others cause rotation to the left, and are called levogymte.

Many other substances, including some liquids and solutions,
also possess this property, though to a much less degree. Thus,
while a plate of quartz 1 mm thick rotates the plane of red light
nearly 18°, the same thickness of turpentine produces a rotation of
only a quarter of a degree.

The method of determining this angle is extremely simple. So-
dium light is commonly used, and the polarizer and analyzer are

THE SACCHARIMETER.

415

so adjusted that the field is dark. The substance to be examined
is then inserted between them, and the analyzer is turned till the
field is again dark ; the angle thus turned through is the rotation
produced by the substance. Liquids and vapors are studied by
enclosing them in a tube of known length through which the polar-
ized light is passed.

These phenomena are explained by supposing that when a beam
of plane polarized light is incident upon the quartz plate it is
doubly refracted, or divided, not into two plane-polarized beams,
but into two circularly-polarized beams, the directions of motion
being opposite. If now these traverse the crystal with slightly
different speeds, it is clear that when, on emerging, they are com-
pounded, the resultant plane of motion and hence plane of polari-
zation will not be parallel to that of the incident light. Evidently
it will have been turned through an angle depending on the thick-
ness of the crystal and on the difference of speeds of the two beams
in it.

In Fig. 319 is shown the tourmaline tongs, a simple form of po-
lariscope. Two plates of tourmaline, cut as described in § 365,
are mounted so as to turn in eyes formed at the extremities of the
looped wire. If the plates be so arranged that the light is com-

FiG. 319.

pletely extinguished, and a pebble (quartz) spectacle lens be placed
between the tourmalines thus arranged, the light will again pass,
showing the effects of rotation of the plane of polarization. This
is accepted as a test of the genuineness of quartz lenses.

375. The saccharimeter.

The angle of rotation of a saccharine solution varies with the
number of grams of sugar in a cubic centimeter of the solution.
On this principle is constructed a polariscope, called a saccharimeter,
which is used for the express purpose of determining the percentage

416 ETHER DYNAMICS.

of pure sugar in an aqueous solution, and hence the commercial
value of a syrup. This instrument is furnished with a scale em-
pirically graduated so that the percentage can be read directly, or
easily calculated.

Section XII.

THERMAL EFFECTS OF RADIATION.

376. Heat not transmitted by radiation. — We have learned
that heat may travel through matter (by conduction), and
with matter (by convection), and it is sometimes stated that
there is a third method by which it travels, viz. " radiation."
Heat itself is not transferred by radiation at all ; heat gen-
erates radiation (i.e. ether waves) at one place, and radiation
produces heat at another; it is radiation which travels, not
heat. It does not exist as heat in the intervening space, and
therefore does not necessarily heat the substance filling that
space. Heat can flow only one way, viz. from a given point to
a point that is colder ; radiation travels in all directions. The
sun sends us no heat, but it sends radiations which the earth
transforms into heat ; but it should be borne in mind that
while it is radiation it is not heat, and vice versa. Tempera-
ture is a condition of bodies, not of radiations ; wave-lengths
belong to radiations, not to heat which produces them.

377. Diathermancy and athermancy. — What becomes of
radiations which strike a body depends largely upon the
character of the body. If the nature of the body be such
that its molecules can accept the motion of the ether, the
vibrations of ether are said to be absorbed by the body and
the body is thereby heated, i.e. the undulations of ether are
transformed into molecular energy or heat. Glass, for in-
stance, allows the sun's radiations to pass very freely through
it, and very little is transformed into heat. But if the glass
be covered with the soot of a candle flame, the soot will

DIATHERMANCY AND ATHERMANCY. 417

absorb the radiations and the glass become heated. Observe
how Gohl window-glass may remain, while radiations pour
through it and heat objects in the room. Only those radia-
tions that a body absorbs heat it ; those that j^ccss through it do
not affect its temperature.

Bodies that transmit radiations freely are said to be dia-
thermanous, while those that absorb them largely are called
athermanous. These terms bear the same relation to the
transmission of radiant energy of any and all wave-lengths as
do transparency and opacity to the transmission of light or
visible radiations. The most diathermanous substance known
is rock salt. A solution of iodine in carbon bisulphide
absorbs almost completely the rays of the visible spectrum,
but transmits almost completely all of longer wave-length
than the red end of the spectrum. A plate of alum acts in
the reverse manner, transmitting the visible and absorbing
the invisible. Among liquids carbon bisulphide is exception-
ally transparent to all forms of radiation ; while water, trans-
parent to short waves, absorbs the longer waves, and is thus
quite athermanous.

Experiment 1. — Bring the bulb of an air thermometer into the focus
of a burning-glass exposed to the sun's rays. The radiation concentrated
on the enclosed air scarcely affects this delicate instrument.

Experiment 2. — Cover the outside of the bulb of the air thermometer
with lamp-black and repeat the last experiment. The lamp-black
absorbs the radiant energy, and the heat conducted through the glass to
the enclosed air raises its temperature and causes it to expand and
rapidly push the liquid out of the stem.

Dry air is almost perfectly diathermanous. All of the
sun's radiations that reach the earth pass through the atmos-
phere, which contains a vast amount of aqueous vapor. This
vapor, like water, is comparatively opaque to long waves :
hence it modifies very much the character of the radiations
which reach the earth. This fact, together with what we have

418

ETHER DYNAMICS.

learned from Exp. 1, enables ns to understand the method by
which our atmosphere becomes heated. Eirst, that portion
of the radiant energy which comes to us from the sun in the
form of relatively long waves is stopped by the watery vapor
in the air^ which is thereby heated. The portion that comes to
us in short waves escaping this absorption heats the earth by
falling upon it. The warmed earth loses its heat, — partly by
conduction to the air, still more largely by radiation out-
ward. The form of radiation, however, has been greatly
changed; for now, coming from a body at a low temperature,
it is chiefly in long waves that the energy is transmitted ;
while, as we have seen, it was largely in the form of short
waves that the earth received its heat. But it is exactly
these long waves which are most readily stopped by the atmos-
phere ; hence, the atmosphere, or rather the aqueous vapor
of the atmosphere, acts as a sort of trap for the energy which
comes to us from the sun.

Kemove the watery vapor (which serves as a " blanket '^ to
the earth) from our atmosphere, and the chill
resulting from the rapid escape of heat by
radiation would probably put an end to all
animal and vegetable life. Glass does not
screen us from the sun's heat, but it can very
effectually screen us from the heat radiated
from a stove or any other terrestrial object.
Glass is diathermanous to the sun's radiations
(simply because they have already lost most
of the very long waves by atmospheric ab-
sorption), but quite athermanous to other
radiations. This is well illustrated in the
case of hot-beds and green-houses. The
sun's rays pass through the glass of these
enclosures almost unobstructed, and heat the
Fig. 320. earth ; but the radiations given out in turn

THE RADIOMETER. 419

by the earth are such as cannot pass out through the glass,
and hence the heat is retained within the enclosures.

378. The ixuliometer.

Fig. 320 represents an instrument called a radiometer. The
moving part is a small vane resting on the point of a needle. It is
so nicely poised on this pivot that it rotates with the greatest free-
dom. To the extremities of each of the four arms of the vane are
attached disks of aluminum or mica which are light on one side
and black on the other. The whole is enclosed in a glass bulb from
which the air is nearly exhausted, i

Exposed to the radiations of the sun, a candle flame, or even the
radiations from the human body, the vane will rotate with the un-
blackened faces in advance. The blackened faces absorb the
radiant energy and become heated, the air particles remaining in the
bulb by striking against them have their speed increased, and thus
results an increased pressure upon the blackened surfaces, causing
a more or less rapid revolution of the arms.

It can be shown that the glass bulb tends to rotate in the oppo-
site direction to that of the vane. This is a proof that the rotation
is due to action and reaction between the vane and the glass, and
not, as it might appear, between the vane and the source of radia-
tion. The radiometer thus serves indirectly to transform radiant
energy into mechanical work, and may be used to measure the
mechanical effects of radiant energy. For example, the radiations
. from two candle flames produce twice the effect of that from one ;
and when the distance from the source of radiations is doubled the
effect is one-fourth as great.^ Hence the radiometer may be used
to verify in a very direct and simple manner the law of inverse
squares as applied to radiant energy. It is also used in a variety
of experiments to illustrate the mechanical effects of the rapidly
moving molecules of gases.

If the opening at the top be kept open and connected with a
pump, so that the exhaustion can be regulated at will, then after
a certain degree of exhaustion has been attained the black disks
exposed to radiation are repelled and rotation ensues. If the

1 The bulb must be exhausted of air till the mean free path of the air particles is
greater than the distance of the glass from the surface of the vane, so that particles
after impinging on the disks do not as a rule collide with other particles before
reaching the glass.

420

ETHER DYNAMICS.

exhaustion be gradually increased, a maximum speed is reached ;
further exhaustion diminishes the speed, and ultimately rotation
ceases.

379. Solar radiation. Pyrheliometer.

Fig, 321 represents an instrument, called a pyrheliometer^ used
to determine solar radiation. It consists of a shallow cylindrical
vessel, A, of thin metal. The upper surface is covered with lamp
black. The bulb of a thermometer is
enclosed in this cylinder and its stem in
a tube, B ; the remaining space in cylin-
der and tube is filled with water. The
blackened surface is turned toward the
sun, and to ensure that the rays are
normal to this surface the shadow of the
cylinder is made to cover exactly and
coincide with the disk C.

First the instrument, sheltered from
the sun, is permitted to radiate its heat
into the clear sky for (say) five minutes.
Let the fall in temperature be r°.

Next it is turned to the sun for five
minutes. Let the rise in temperature be
E°.

Finally it is allowed {at its increased
temperature) to radiate into the clear sky
as before for five minutes. Let the fall of temperature be r'°.

Now since r denotes the change of temperature during radiation
into clear sky before heating, ^d r' tho same after heating, the
change of temperature during the heating, due to radiation, con-

r -\- r'
duction, etc., will be very nearly a mean between the two, or — - — ;

but it is evident that this cooling effect takes place even when the
instrument is receiving the sun's radiations, and tends to diminish
the heating effect produced by these radiations. Hence, therefore,
the whole heating effect will be

r -\- r'

Fig. 321.

This is the number of degrees which the sun's radiant energy is
able to raise the temperature of n kilograms of water (suitable al-

ALL BODIEvS RADIATE HEAT. 421

lowance being made for the water equivalent of the vessel and
thermometer) when it falls upon a units of area of lamp black for
m units of time. Then the radiant energy received per unit area
during one unit of time is equivalent to

(«

7^ -I- y' ~

H -— ) ^ (a X m)

calories of heat. By Joule's equivalent we are able to calculate the
energy in mechanical units.

It is estimated that if the whole radiation received by the earth
from the sun were employed in melting ice, it would in a year melt
a layer of ice all round the earth 137 feet in thickness ( Young). i
Only a very small fraction of the sun's radiations are intercepted
by the earth. Lord Kelvin estimates that the total energy emitted
by the sun is at the rate of 7000 horse-power per square foot of
radiating surface.

380. All bodies radiate heat. — Hot bodies usually part with
their heat much more rapidly by radiation than by all other
processes combined. But cold bodies, like ice, emit radia-
tions even when surrounded by warm bodies. This must be
so from the nature of the case, for the molecules of the coldest
bodies possess some motion, and being surrounded by ether
they cannot move without imjjarting some of their motion to
the ether, and to that extent becoming themselves colder.

381. Prevosfs theory of exchanges. — Let us suppose that
we have two bodies, A and B, at different temperatures, — A
warmer than B. Kadiation takes place not only from A to B,
but from B to A ; but, in consequence of A's excess of tem-
perature, more radiation passes from A to B than from B to
A, and this continues until both bodies acquire the same
temperature. At this point radiation by no means ceases,
but each now gives as much as it receives, and thus equilib-
rium is kept up. This is known as "Prevost's Theory of
Exchanges."

1 For further information concerning solar radiation, see Young's Elements of
Astronomy, pp. 147-153.

422 ETHER DYNAMICS.

382. Good absorbers, good radiators.

Experiment 3. — Select two small tin boxes of equal capacity, — one
should be bright outside, while the other should be covered thinly with
soot from a candle flame. Cut a hole in the cover of each box large
enough to admit the bulb of a thermometer. Fill both boxes with hot
water, and introduce into each a thermometer. They will register the
same temperature at first. Set both in a cool place, and in half an hour
you will find that the thermometer in the blackened box registers several
degrees lower than the other. Then fill both with cold water, and set
them in front of a fire or in the sunshine, and it will be found that the
temperature in the blackened box rises more rapidly.

As bodies differ widely in their absorbing power, so they
do in their radiating power, and it is found to be universally
true that good absorbers are good radiators, and bad absorbers
are bad radiatoi^s. Much, in both cases, depends upon the
character of the surface as well as of the substance. Bright,
polished surfaces are poor absorbers and poor radiators ;
while tarnished, dark, and roughened surfaces absorb and
radiate rapidly. Dark clothing absorbs and radiates more
rapidly than light clothing.

383. Deta. ■ — It requires no elaborate experiments to show
that some bodies radiate more rapidly than others. All
nature testifies to this, every still, cloudless summer night.
During the day objects on the earth's surface gain more heat
by radiation than they lose, but as soon as the sun has set
this is reversed. Then everything begins to cool by radia-
tion into space. Objects becoming cool, the air in contact
with them becomes chilled ; its watery vapor condenses, and
collects in tiny liquid drops on their surfaces. But these
dew-drops collect much more abundantly on certain things,
such as grasses and leaves, than on others, such as stones and
earth. The reason that dew does not collect on the latter so
freely, is because of their poor radiating power ; they do not
get cool as rapidly.

QUESTIONS. 423

Questions.

1. What objections can you raise to the term " radiant heat " ?

2. Explain why the temperature of a hotbed is above that of the sur-
rounding air,

3. How could you separate the dark radiation of an electric arc lamp
from tlie luminous radiation ?

4. How can you demonstrate the existence of ether waves of greater
length than the light-giving waves ?

5. Ice appears to radiate cold. Explain the phenomenon by Provost's
theory.

6. A radiometer placed near ice rotates with the blackened sides of its
disks in advance. Explain.

7. What parts of the spectrum are invisible to the eye ?

8. How can you prove tlie existence of invisible solar rays ?

9. On what does the color of bodies primarily depend ?

10. What agency does a body perform in determining its own color
when illuminated with white light ?

11. a. Why is grass green ? b. Snow, white? c. Soot, black?

12. How does a spectrum produced by a crown glass prism differ from
a spectrum produced by a flint glass prism ?

13. State some phenomenon which the undulatory theory alone is
competent to explain.

14. Describe the appearance which an iridescent soap bubble would
present in a monochromatic light.

15. Objects seen across the top of a hot stove appear unsteady and
indistinct. Explain.

16. State how with a pair of tourmaline tongs you may distinguish a
glass spectacle lens from a quartz lens.

17. State how the sensation of purple is produced.

18. What is meant by the artist's three primary pigments ?

19. Describe the surface which a hot-water vessel should have in order
to retain its heat well.

20. Suppose beams of sunlight enter a dark room through two aper-
tures in a shutter, and a blue glass be placed in the path of one beam
and a yellow glass in the path of the other, a. What color will that
portion of a white wall appear where the two images of the sun over-
lap ? b. State the result if one of the apertures be closed and the beam
from the other aperture be made to pass through both colored glasses.

21. When red and green sensations coexist what is the resulting sen-
sation ?

424

ETHER DYNAMICS.

22. What phenomenon shows that ether- waves do not traverse all
substances with equal speed ?

23. State how light may "turn a corner."

24. What utility is there in keeping certain parts of a steam engine
very bright ?

Section XIII.

SOME OPTICAL INSTRUMENTS.

384. Coiwpoiind microscope. — When it is desired to magnify
an object more than can be done conveniently and v^ith
distinctness by a single lens, two convex lenses are used, —
one, (Fig. 322), called the objective, to form a magnified
real image a' V of the object a h ; and the other, E, called the
eye-jjiece, to magnify this image so that the image a' b' appears
of the size a" b" . Instead of looking at the object as when
we use a simple lens, we look at the real inverted image, a^b',
of the object.

This represents the simplest
b" possible form of the compound micro-

,/| scope. In practice, however, the

^,;;''' i construction is more complicated.

Fig. 323 represents a perspective
and a sectional view of a simple form
of a modern compound microscope.
The body of the instrument consists
of a series of brass tubes movable one
within another. In the upper end H
is the ocular or eye piece. It consists
of two plano-convex lenses o and n,
the former called the eye-lens, the
latter called the field lens. The ad-
use of two lenses in the eye-piece

Fig. 322.

vantages derived from the
are as follows:

1. The combination diminishes spherical aberration and thereby
increases the flatness of the field. The images a' b' and a'' b" (Fig. 323)
are in reality curved in consequence of the spherical aberration

COMPOUND MICROSCOPE.

425

caused by the objective. The effect of the field lens is to correct
this curvature in a measure.

2. The combination increases the field of view, so that a larger
area of the object is made visible at the same view.

3. The combination diminishes chromatic aberration.

All microscopes, however, should be furnished with an achro-

FlG. 323.

matic objective. This consists of two to four achromatic lenses,
(the achromatic triplet, the most common form, is represented on
an enlarged scale at L in Fig. 323), combined so as to act as a
single lens of short focus. By the use of several lenses, the aber-
rations can be better corrected than with a sinsrle lens.

426

ETHER DYNAMICS.

The object to be examined is placed on a stage, S, and, if
the object be transparent, it is strongly illnminated by focus-
ing light upon it by means of a concave mirror, M. If the
object be opaque, it is illuminated by light directed upon it
obliquely from above by the converging lens N.

385. Oculars.

The negative (or Huyghenian) ocular consists of two convex
lenses of crown glass, F and E (Fig. 324), the convex surfaces
being turned toward the object glass. A pencil of rays from

the object-glass converg-
ing towards a focus, a, is
brought to a focus, a\
half way between the two
lenses.

This ocular is called
negative because it is adapted to rays already converging. The
focal length of F is three times that of E, and the distance between
the lenses is one-half the sum of the focal lengths.

The positive (or Ramsden) ocular consists of two plano-convex
lenses, E and E' (Fig. 325), with the convex surfaces turned
towards each other. These lenses are of equal focal length, and
the distance between
them is two-thirds the
focal length of one of
them. This combi-
nation is not achro-
matic. It is used

Fig. 324.

Fig. 325.

when spider lines are placed in the focus of the field lens for pur-
poses of exact measurement.

In obtaining high magnifying power, it is generally best to use
objectives of short focal length rather than oculars of high power,
as the latter magnify the imperfections of the former.

386. Magnifying power.- — The magnifying power of a com-
pound microscope is the product of the respective magnifying
powers of the object-glass and the eye-piece; that is, if the
first magnify 20 times and the other ten times, the total
magnifying power is 200. The magnifying power is deter-

TELESCOPES. 427

mined experimentally by means of a micrometer scale, for a
description of which the student is referred to technical works
on microscopy.

387. Telescopes. — Telescopes are used to view (scope) ob-
jects afar off (tele). They are classified as astronomical or
terrestrial, according as they are designed to be used in view-
ing heavenly bodies or terrestrial objects ; reflecting or re-
fracting, according as the objective is a concave mirror or a
converging lens. The terrestrial telescope differs from the
astronomical in producing images in their true position with-
out inversion. This is effected by means of an extra object
lens, which corrects the inversion of the main object lens.
The matter of inversion is of little or no consequence in
viewing heavenly bodies.

The refracting astronomical telescope consists essentially,

Fig. 326.

like the compound microscope, of two lenses. The object-
glass (0, Fig. 326) forms a real diminished image (a h) of the
object A B ; this image, seen through the eye-glass E, ax3pears
magnified and of the size c d. The object-glass is of large
diameter, in order to collect as much light as possible from a
distant object for a better illumination of the image.

This telescope is analogous to the microscope, but the two in-
struments differ in this respect : in the microscope, the object being
very near the object-glass, the image is formed much beyond the
principal focus, and is greatly magnified, so that both the object-
glass and the eye-piece magnify ; while in the telescope, the heavenly
body being at a great distance, the incident rays are practically

428

ETHER DYNAMICS.

parallel, and the image formed by the object-glass is much smaller
than the object. The only magnification which can occur is pro-
duced by the eye-piece, which ought therefore to be of high power.
The magnifying power of this instrument i equals approximately
the focal length of the object-glass divided by the focal length of the
eye-piece.

388. The Newtonian reflecting telescope.

For an instrument of moderate cost, specially adapted to school
and college use owing to the ease of manipulation and the comfort
with which the observer may view any part of the sky, the New-
tonian reflecting telescope meets with much favor. It also possesses
the great advantage of giving a colorless image of bright objects,

Fig. 327.

which cannot be obtained in a refractor. 2 Fig. 327 represents a
horizontal sectional view of this instrument. Incident rays are
reflected from the parabolic mirror, M ; striking the rectangular
prism, m n, they undergo total reflection, and form at a 6 a small
image of the heavenly body. The image is viewed through an eye-
piece inserted in the side of the telescope. The reflector serves as
an object-glass, and is of course free from chromatic aberration,
while spherical aberration is corrected by the shape given the

1 The student may ascertain the magnifying power of a terrestrial telescope by
viewing a scale directly with one eye, and its magnified image as seen through the
telescope with the other eye. (See the author's Laboratory Manual and Note Book.)

2 On the whole, however, if the matter of expense be disregarded, the balance of
advantage is generally considered to lie with the refracting telescope. Briefly, its
chief advantages are : (1) it gives a brighter image than a reflector of the same size ;
(2) it gives a better definition ; (3) the lens does not deteriorate with age as does the
speculum in a reflecting telescope.

The largest refracting telescope that has thus far been made (except the Yerkes
telescope at the University of Chicago) is that of the Lick Observatory in California
(see Plate in).

Plate HL

PHOTOGRAPHER S CAMERA.

429

389. Photograypher^ s camera. — The jjJiotographer^s camera
or camera ohscura, of which A B (Fig. 328) represents a ver-
tical section, consists of a dark box painted black on the

Fig. 328.

interior. A screen of ground glass, S, forms a partition in
the box. A sliding tube, T, contains a convex lens, L. If
an object, D, be placed some distance in front, and the dis-
tance of the lens from the screen be suitably adjusted, a
distinct, real, and inverted image can be seen upon the screen
by looking through the aperture C. When the image is prop-
erly focused, the photographer replaces the ground-glass plate
by a sensitized plate, and by their chemical power the sun's
rays imprint a true picture of the object on this plate.

390. The human eye. — Fig. 329 represents a horizontal
section of this wonderful organ,
eye, like a watch-crystal, is a
transparent coat 1, called the
cornea. A tough membrane 2,
of which the cornea is a con-
tinuation, forms the outer
wall of the eye, and is called
the sclerotic coat, or "white
of the eye." This coat is
lined on the interior with
a delicate membrane 3, called
the choroid coat ; the latter con-
sists of a black pigment, which fig. 329.

Covering the front of the

430 ETHER DYNAMICS.

prevents internal reflection. The inmost coat 4, called the
retina, is formed by expansion of the optic nerve 0. The
muscular tissue ii is called the iris; its color determines
the so-called " color of the eye." In the center of the iris is
a circular opening 5, called the pu23il, whose function is to
regulate, by involuntary enlargement and contraction, the
quantity of light-waves admitted to the posterior chamber of
the eye. Just back of the iris is a tough, elastic, and trans-
parent body 6, called the crystalline lens. This lens divides
the eye into two chambers ; -the anterior chamber 7 is filled
with a limpid liquid, called the aqueous humor ; the posterior
chamber 8 is filled with a jelly-like substance, called the
vitreous humor. The lens and the two humors constitute the
refracting apparatus.

Experiment 1. — a. Make a model of an eye. Fill an 8-ounce flask
with clear water (eye-ball). Cover one side with black paper having a
round hole in it (iris and pupil). Place a slightly convex lens in front of
the hole (cornea and crystalline lens combined ; the latter outside the
eye-ball instead of inside). Place a candle flame (object) in front of the
hole at a distance of about 4 feet ; catch (inverted) distinct image of the
flame on a paper screen (retina) behind the flask. Move the candle nearer
the flask ; the image becomes indistinct. Eestore distinctness by inter-
posing a converging lens (remedy for long sight).

6. Place the candle very near the lens and focus its image on the
screen (now in a new position). Move the candle away ; the image comes
nearer the lens, and to carry the image back to the screen you must use
a diverging lens (remedy for short sight).

Experiment 2. — Make two dots on paper two inches apart. Close
the left eye, and bring the right one over the left spot. At a distance of
about six inches the right spot becomes invisible. As you bring the
paper nearer, the eye turns to regard the left spot ; the image of the right
spot meantime travels noseward over the retina, until it reaches a spot
on the retina, called the blind spot, which is not sensitive to the action
of light-waves. This spot is where the optic nerve enters the eye.

The eye is a camera obscura, in which the retina serves as
a screen. Images of outside objects are projected by means

DEFECTS OF VISION. 431

•of the crystalline lens^ assisted by the refraction of the
humors, upon this screen, and the impressions thereby
made on this delicate network of nerve filaments are conveyed
by the optic nerve to the brain. If the two outer coatings be
removed from the back part of the eye of an ox recently
killed, so as to render it somewhat transparent, true images
of whole landscapes may be seen formed upon the retina of
the eye, when it is held in front of your eye.

With the ordinary camera, the distance of the lens from
the screen must be regulated to adapt itself to the varying
distances of outside objects, in order that the images may be
properly focused on the screen. In the eye this is accom-
plished by changing the convexity of the lens. We can almost
instantly and unconsciously change the lens of the eye, so as
to form on the retina a distinct image of an object miles away
or only a few inches distant. The nearest limit at which an
object can be placed so as to form a distinct image on the
retina is about five inches. On the other hand, the normal
eye in a passive state is adjusted for objects at an infinite
distance.

The retina, on careful examination, is found to be com-
posed in part of little elements in its back portion, which
have received, from their appearance, the names of rods and
cones. It is thought that these rods and cones receive and
respond to the vibrations of ether ; in other words, that they
co-vibrate with the undulations of the ether, and thereby we
get our sensation of light.

The eye is not free from spherical or chromatic aberration,
though these are very much reduced by the action of the iris,
which acts as a diaphragm to cut off all except the central
rays.

391. Defects of vision. — Myopia (short-sightedness) is caused
by the excessive length of the globe from front to back, so
that the images of all but near objects are formed in front of

432

ETHER DYNAMICS.

the retina. Eemedy : use diverging lenses. Hypermetropia
(long-sightedness) occurs when the axis of the globe is so short
that the image of an object is back of the retina unless the
object is held at an inconvenient distance, in which case it
tends to become indistinct. Eemedy : use converging lenses.
Presbyopia is due to loss of accommodation power, so that while
vision for distant objects remains clear, that for near objects
is indistinct. This defect is incident to advancing years, and
is due to progressive loss of elasticity of the crystalline lens.
Eemedy : converging lenses. Astigmatism is caused by an
inequality in the curvature of the cornea in different meridi-
ans, so* that when, for example, a diagram like Fig. 330 is
held at a distance, vertical lines will
be in- focus and horizontal lines will be
out of focus and will appear blurred
and indistinct, or vice versa. Eemedy:
lenses of cylindrical curvature. But,
for this, as well as for all other de-
fects or troubles of the eyes, consult
a skilled oculist, and the earlier the
better.

Advice to all : Do not overstrain or
overtax the eyes, or use them in in-
sufficient or excessive light, in flickering light such as that of
a gas-jet, or in unsteady light such as that in a moving
vehicle ; and avoid so far as practicable sudden changes of
light, such as lightning flashes, etc.

392. Stereo2oticon. — This instrument is extensively em-
ployed in the lecture-room for producing on a screen magnified
images of small, transparent pictures on glass, called slides;
also for rendering a certain class of experiments visible to . a
large audience by projecting them on a screen.^ The lime

iFor useful information relating to the operation of projection, especially for
scientific illustrations, see Wright's Light, and Dolhear's Art of Projecting.

Fig. 330.

STEREOPTICON.

433

light is most commonly used, though the electric light is pre-
ferred for a certain class of projections. The flame of an
oxyhydrogen blow-pipe A (Fig. 331) is directed against a
stick of lime B, and raises it to a white heat. The radiations

Fig. 331.

from the lime are condensed, by means of a convex lens c,
called the condensing lens, (usually two plano-convex lenses
are used), so that a larger quantity of radiations will pass
through the convex lens E, called the projecting lens. The
latter lens produces (or projects) a real, inverted, and mag-
nified image of the picture on the screen S. The mounted
lens E may slide back and forth on the bar E, so as properly
to focus the image.

434

ETHER DYNAMICS.

CHAPTER II.
ENERGY OF ETHER-STRAIN. —ELECTRO-STATICS.
Section I.

INTRODUCTION.

393. Electrification. — Certain bodies, when the conditions
are suitable, acquire by contact and subsequent separation (or
more readily by friction^) the property of attracting light
bodies such as feathers, pieces of tissue paper, etc. For
example, glass rubbed with silk, and sealing-wax or ebonite
with woolen cloth, manifest this property by picking up
scraps of paper, etc. Bodies in this state are said to he
electrified or charged with electricity.

Experiment 1. — Balance a flat wooden ruler (Fig. 332) upon the bottom
of an inverted flask. Rub a rubber comb with a woolen cloth or draw

it a few times through your hair (if dry)
and place it near one end of the ruler ;
the ruler will turn toward the comb.

Experiment 2. — Hold the comb over a
handful of bits of tissue paper ; the papers
quickly jump to the comb, stick to it for
an instant, and then leap energetically
from the comb. The papers are first
attracted to the comb, but in a short
time acquire some of its electrification,
and then are repelled. If the papers be
pulled off from the comb, they will cling
Fig. 332. to the hands of the operator.

1 Possibly because in the act of rubbing more points are brought in contact.

TWO KINDS OF ELECTRIFICATION.

435

Fig. 333.

394. Tiuo kinds of electrification.

Experiment 3. — Suspend a ball of elder pith, C (Fig. 333), by a silk
thread. Electrify a glass rod D with a silk handkerchief and present it to
the ball ; attraction at first
occurs, followed by repulsion
soon after contact. Next ex-
cite a stick of sealing-wax or
a rubber comb with a woolen
cloth and present it to the ball
which is repelled by the elec-
trified glass ; it is attracted by
the electrified wax or rubber.

Experiment 4. — Suspend
in two stirrups two glass rods
that have each been rubbed
with silk (Fig. 334), and pre-
sent them to each other; they
repel each other. Suspend two sticks of sealing-wax that have been rubbed
with flannel in the same manner; the same result follows. Now, in a

like manner, present one of the glass rods
and one of the sticks of sealing-wax to
each other ; they attract each other.

It is evident (1) that there are
tiuo kinds or conditions of electrifi-
cation ; or, for convenience, we
sometimes say tiuo kiiids of elec-
tricity ; (2) that bodies similarly
electrified repel one another, bodies oppositely electrified attract
one another.

Glass rubbed with silk is said to receive a charge of vitreous
electrification ; the wax, after being rubbed with woolen cloth,
on the other hand, is charged with resinous electrification.
Vitreous and resinous electrifications bear to each other
somewhat the same relation as positive and negative quanti-
ties in algebra ; and by arbitrary convention vitreous charges
are said to be positive (written +E), and resinous negative
(written — E).

Fig. 334.

436 ETHER DYNAMICS.

Experiment 5. — Once more electrify a stick of sealing-wax with woolen
cloth, and present it to the pith ball, and after the ball is repelled, bring
the surface of the flannel which had electrified the rod near the ball ; the
ball is attracted by it, showing that the rubber is also electrified, and
with the opposite kind to that which the sealing-wax possesses.

One kind of electrification is never developed alone ; when
two substances are rubbed together, and one becomes electri-
fied, electrification of the opposite kind is always developed
in the other.

395. Electric attraction and repulsion explai7iable 07i the
hypothesis of ether-strain. — When small pieces of glass and
silk are rubbed together, it is found that after they are pulled
apart they attract each other w^ith a definite and measurable
force ; and that this force varies inversely as the square of
the distance between them. When two bodies are pulled apart,
energy is expended upon them which will be restored when
they are allowed to approach each other. This phenomenon
is explainable on the hypothesis that in the work of separa-
tion, the ether between or around them is strained ; and that
the tendency of the two bodies to approach each other is the
tendency of the elastic ether to recover its normal condition.
The phenomena of electric attraction and repulsion are most
satisfactorily explained on the hypothesis of ether-strain. By
whatever hypothesis explained, it is certain that electrification
is the result of work done, and is a form of potential energy.

396. What is electricity ? — The student naturally has al-
ready begun to ask the never-answered question, "What is
electricity ? " and to inquire, " What is the function of electri-
city in these operations ? " Provisionally we shall regard elec-
tricity as that which is transferred from one body to another
body when the two become oppositely electrified.^ Electricity

iWhat the ultimate nature of electricity is, whether it he the ether itself or
(more pi'obably) a constituent of the ether "as water is a constituent of jelly " ;
whether it he a fluid (it certainly possesses the property of fluidity) ; whether, accord-
ing to Franklin, a positive charge is an excess and a negative charge a deficit in a

QUANTITY OF ELECTRICITY. 437

is 7iot a form of energy.^ It is quite true that electricity
imder pressure or in inotion possesses energy ; in the same
sense do water and air under like conditions possess energy,
but we do not therefore deny them to be forms of matter.
Electricity, rather, in many respects possesses the nature of
matter. Like matter it can neither be created nor annihil-
ated, and like matter it can be moved and put under stress.
For present purposes, then, electrification may be regarded as
a state of strain in some intervening medium produced by a
transfer of electricity from one body to another and the sub-
sequent separation of the two bodies. Electrification is the
result of work done, and is most certainly a form of energy.

397. Quantity of electricity.

When we do not know what a thing is, it is difficult to conceive
a definite quantity of it. But our knowledge of electricity like that
of force is derived from its effects. From the measurement of its
effects, therefore, can we define a unit quantity of electricity. For
purposes of calculation at least, electricity of either kind may be
treated precisely as if it were a material incompressible fluid, and
any increase or decrease of electrification may be considered to be
produced by the addition or taking away of a quantity of electricity.
Quantities of electricity are added and subtracted by the usual rules
of algebra, the kinds of electrification being denoted by the + and
— signs.

398. Latv of attraction.

When two equally electrified bodies attract or repel each other
with a force of one dyne at a distance of one centimeter in air, each
is charged with a certain definite quantity which may be taken as
a unit quantity of electricity. If two bodies repel or attract each

certain standard quantity of the fluid which all bodies are supposed to possess in
their unexcited state ; or whether (more probably) positive and negative electricities
are distinct entities (whose relations to each other are more like those of sodium
and chlorine than like those of heat and cold), such that when combined in any
body they neutralize each other so that the body possesses none of the properties of
electrification, but when they by any means become separated two separate portions
of matter always become oppositely charged — are questions too recondite for dis-
cussion in any general work on physics. Consult " Modern Views of Electricity,"
by O. J. Lodge.

1 See Daniell's Principles of Physics, p. 530.

438

ETHER DYNAMICS.

other with some other intensity, the quantities with which they are
charged are easily determined. For example, suppose that a body
charged with three units is attracted at a distance of one centimeter
by one charged with six units. The tol^l attraction of the six units
of the second body for each one of the other three is obviously
expressed by six, giving a total attraction of 6 X 3 = 18 dynes.
It is evident that if any two of these three quantities be known,
the third can be determined.

Again, suppose that the attraction (or repulsion) be at some other
distance, the force, being a radiant one, varies inversely as the square
of the distance ; consequently, to determine this force the product of
the two quantities must be divided by the square of the distance
between them. Now if we substitute quantities of electricity, q and
g', for masses m and mf in the formula for attraction of gravitation
(§ 96), we shall have the formula for electrical attraction (or re-
pulsion) A ; viz. A = z—r- , in which k is the dielectric constant

(see § 409). That is, the electrical attraction between two charged
bodies (provided the areas of the bodies are small so as to keep
them under the law of radiant force) varies as the products of their
charges and inversely as the square of the distance between them.

399. Electroscope. — This is an instrument used to detect
the presence of electrification in a body, and to determine its
kind. It usually consists of two strips of gold foil, A B (Fig.

335), suspended from a brass
rod within a glass jar. To
the upper end of the rod is
fixed a metal disk, C. On
the opposite sides of the in-
terior of the jar are two strips
of metal foil, D and E, of suf-
ficient hight to be touched by
the strips A and B on their
extreme divergence.

(1) If an unelectrified body

be brought near the disk C,

Fig. 335. no ciiange takes place in the

CONDUCTION. 439

two strips of foil A and B^ but if an electrified body be
brought near the disk, the strips diverge, thus indicating the
existence of a charge of electricity in the body.

(2) If the electroscope be charged by contact with an ex-
cited body, the strips will remain in a divergent position.
While in this condition, if a body similarly charged be
brought near the disk, the strips will diverge more; but if
an unexcited body, or a body oppositely electrified be brought
near the disk, the strips will collapse.

400. Conduction.

Experiment 6. — a. Eub a brass tube, held in the hand, with warm
silk. Bring it near the disk of the electroscope ; the leaves are unaffected.
6. Wrap a piece of sheet rubber around one end of the tube and hold this
end in the hand, and rub as before. Bring it near the disk of the elec-
troscope ; notice that the leaves diverge, c. Repeat the last operation ;
but before bringing the tube near the disk touch the tube with a finger.
The leaves no longer show signs of electrification.

In the first (a) and last (c) operations electricity escaped
through the hand and body to the earth ; in the second (h) it
was prevented from escaping by the intervening sheet rubber.
Substances which allow electricity to spread over them, i.e.
substances which offer little ^resistance to the flow of elec-
tricity, are called conductors. Those which offer great resist-
ance to its passage are called non-conductors, insulators, or
dielectrics.

Some of the best insulating substances are dry air, ebonite,
shellac, resins, glass (free from lead, e.g. common bottle glass),
silks, and furs. On the other hand, metals are, as a class,
exceedingly good conductors. Moisture injures the insulation
of bodies ; hence experiments succeed best on dry, cold days
of winter, when moisture of the air is least liable to be con-
densed on the surfaces of apparatus, especially if it he kept
warm.

Water cannot be retained in a reservoir unless its walls be
of sufficient strength ; so a body, in order to become charged

440

ETHER DYNAMICS.

and to retain the charge, must be surrounded by something
that will offer sufficient resistance to the escape of electricity.
As regards a specific body, there is no limit to the quantity of
electricity with which it can be charged, provided the charge
can be retained. This entity which represents the walls of the
reservoir is termed the dielectviG. It may be the air ^ or any
of the so-called non-conductors of electricity. Even the ether
may be considered a dielectric. A body thus surrounded is
said to be insulated.

Experiment 7. — Prepare an insulated stool by placing a board on four
dry and clean glass tumblers, used as legs. Let a person, whom we will
call John, stand on this stool, and hold in one hand one end of a wire
(say) 4 yds. long, the other end of which is attached to the disk of an

_ electroscope. a. Let a

second person, James,
strike John with a cat's
fur ; the leaves diverge. ,
h. Substitute a white silk
thread for the wire, touch
the electroscope with a
finger io as to discharge
it, and repeat the last op-
eration ; the leaves do not
diverge. . c. Let James
strike John several times
with the fur, and then
bring a finger knuckle
near to some part of
John's person, e.g. the
chin, nose, or a knuckle
(Fig. 336); an electric spark will pass between the two and both will
experience a slight shock. The electricity which had accumulated on
John in consequence of his insulation, escapes or is discharged through
James.

Fig. 336.

1 If the air were a conductor of electricity, a body could not be charged in it;
there could be no thunder storms, and man would probably never have known of the
existence of electricity.

ELECTRICITY ACTS ACROSS A DIELECTRIC.

441

Section II.

INDUCTION.

401. Electricity acts across a dielectric.

Experiment 1. — Fig. 337 represents an empty egg-shell covered with
tin foil to make it a good conductor. It is suspended from a glass rod
by a silk thread, a. Electrify a
glass rod and bring it near the
shell. The shell moves toward the
rod. h. Next introduce a glass
plate between the rod and shell.
The shell approaches the rod as
before.

The chief lesson we learn
from this experiment is that
electricity acts across a dielec-

tric.
air :

In a the dielectric was

Fig. 337.

in b, air and glass.
402. To determine what ac-
tually happens on an insulated
conductor when an electrified
body is brought near.

Experiment 2. — a. Suspend, as above, two shells so as to touch each
other, end to end, as in Fig. 338, thus making practically one conductor.
Bring near to one end of the shells a sealing-wax rod, D, excited with
— E. While the rod is in this position carry a thin strip of tissue paper,

C, along the shells. The paper is attracted to the shells, but most
strongly at the ends. In the middle of the conductor, where the shells
touch each other, there is little if any electrification.

6. While the rod D is still in position, separate B from A, then remove

D. Test each shell with the tissue paper ; both are found to be excited,
c. Charge an electroscope with +E. Then bring A near it • the leaves

diverge, showing that A is charged with -HE. Bring B near the electro-
scope ; the leaves collapse, showing that B is charged with — E.

1 Insulators across which electric action takes place are called dielectrics, from
the Greek 5id, across.

442

ETHER DYNAMICS.

d. Finally bring the two shells near each other ; they attract each
other. Allow them to touch each other, and then test each with the
tissue paper or the electroscope ; it will be found that both have become
discharged.

From the above operations we learn that when an electri-
fied body is brought near but not in contact with an insulated

Fig. 338.

conductor, the electrified body acts across the dielectric upon
the conductor, repelling electricity of the same kind to the
remote side of the conductor, and attracting the opposite kind
to the side near to it. Such electrical action is called induc-
tion. The electrified body which produces the action is called
the inducing body ; the charge of electricity thus produced is
called induced electricity.

403. Charging by induction.

Experiment 3. — Take a proof plane E (Fig. 339) (which consists of an
insulating handle of glass or gutta percha, terminating at one end with a
thin metal disk, F, about the size of a 5-cent nickel), and connect it with
an electroscope, G, by a fine wire, H. Bring a stick of sealing-wax
electrified as before with — E near the egg-shell conductor. Holding
the proof plane by the insulating handle, bring the disk near the end of
the conductor charged by induction with — E. The — E will act induc-
tively upon the continuous conductor consisting of disk, wire, and
electroscope, charging the end nearest itself {i.e. the disk) with +E and

CHARGING BY INDUCTION.

443

Fig. 339.

the remote end (i.e. the leaves) with — E. The leaves of the electro-
scope show the presence of a charge by their divergence.

Now while everything is in the position indicated by the cut, touch
with the finger any
part of the continu-
ous conductor ; the
leaves of the electro-
scope instantly col-
lapse. The — E with
which the leaves had
been charged being
free is discharged
through your body.
But the + E concen-
trated on the disk of
the proof plane is

hound by the attraction of the charge of — E on the end of the shell
nearest it, and cannot escape. Kemove the finger from the electroscope
and the proof plane from the influence of the shell; the leaves again diverge.

The last phenomenon' is explained as follows : After — E
had been discharged from the continuous conductor, there
was left an excess of +E; but this excess was all concen-
trated in the disk F so long as it remained near the negative
charge of the shell. But as soon as E was removed from the
influence of the shell, the charge spread itself over the entire
conductor, and the leaves, which received a portion of the
charge, diverged. The conductor is said to be charged by
induction.

Ex'periment 4. — To electrify the shell by induction, bring the excited
wax near it, touch the shell with a finger, remove the finger, and finally
remove the rod. The proof plane being connected with the electroscope
and being charged with — E, bring E near to the shell A ; the leaves
collapse, showing that the shell is charged with +E, which draws the
— E away from the leaves.

Observe that when a body becomes charged by induction
the charge which it receives is opposite in kind to that of the
inducing body.

444 ETHER DYNAMICS.

404. Charging by conduction.

Experiment 5. — Disconnect the proof-plane from the electroscope.
Charge the electroscope with — E and the shell with +E ; touch the
shell with the disk of the proof plane, then hold the disk near the
electroscope ; the divergent leaves collapse, showing that the disk bears
-f-E which it received by conduction from the shell when they were
brought in contact. Of course the charge is the same kind as that of the
body which communicated it.

405. l7iduction precedes attractio7i. — When a pith ball is
brought near an electrified glass rod, the -|- E on the rod A

(Fig. 340) induces — E on the side of the ball B
nearest A and repels +E to the farther side.
The -j-E of A and the — E of B therefore attract
each other ; likewise the +E of A and the +E
of B repel each other ; but since the former
charges are nearer each other than the latter
are, the attraction exceeds the repulsion.
Fir 340 ^^^* Electrification confined to the outside sur-

face of a conductor. Metal screens.

Experiment 6. — Place a tin cup, A (Fig. 341), on a glass tumbler
coated with shellac and charge it heavily with electricity from an electri-
cal machine (see Section V.). Introduce a proof-
plane into the cup and touch the interior surface of
the cup. Remove the proof-plane and place it near
the electroscope ; the leaves of the electroscope are
unaffected.

If a solid metal ball, A (Fig. 342), sus-
pended by an insulating thread, be electrified
and then covered with two hemispherical
metallic cups, B and C, having insulating
handles, and the cups be afterwards re-
moved, the ball when tested with the elec- fig.341.
troscope will be found to have lost all its charge, while the
cups will be found to be charged. It does not make the
slightest difference as to the result whether an insulated

FARADAY S ICE-PAIL EXPERIMENT.

445

Fig. 342.

conductor be solid or hollow. Wood covered with tin-foil
answers the purpose as well as any other body.

If a hollow conductor he
charged, however highly,
with electricity, the whole
of the charge is found
upon the outside surface.
If the electroscope in
the last experiment were
placed inside the tin cup,
or if it be set inside a vessel of wire gauze (e.g. a bird cage),
and the vessel be charged with electricity or a heavily
charged body be brought near the vessel, the electroscope will
be unaffected. This interesting and important fact shows
that a metallic shell, however thin, entirely screens bodies inside
it from external electrification,' however great.^

407. " Faraday'' s Ice-pail Experiment.''^

Experiment 7. — a. Insulate well a tin pail (Eig. 343) and connect it
with an electroscope. Charge heavily a metal ball suspended by an in-
sulating thread, with (say) 4-E. Lower
the ball within the pail ; the pail be-
comes charged by induction, inside
with — E and the outside (together with
the electroscope) with 4-E. The leaves
of the electroscope diverge.

h. Touch the outside of the pail with
a finger ; the free charge of + E escapes
to the earth (see § 413), and the leaves
of the electroscope collapse. Remove
the finger and then withdraw the ball
slowly from the pail ; the leaves of the
electroscope slowly diverge, and remain diverged after the ball is removed.

1 To test this stiU further, Faraday built a cubical cage of 12 ft. edge, of copper
wire, and lined the interior with paper covered with tin-foil. This chamber was in-
sulated and put in connection with a powerful electrical machine while working.
He says : — "I went into the cube and lived in it, using electrometers and all other
tests of electrical states ; I could not find the least influence upon them, though all
the time the outside of the cube was powerfully charged, and large sparks and
brushes were darting from every part of its outer surface."

Fig. 343.

446 ETHER DYNAMICS.

c. The ball is now charged with + E and the pail with — E. Bring
the ball in contact with the pail ; the leaves of the electroscope completely
collapse, showing that the +E and the — E have combined, neutralizing
each other and- leaving no excess of either. Hence we conclude that the
positively charged ball when lowered into the pail must have induced an
equal charge of — E in the pail. This is generally called the Ice-pail
Experiment because Faraday in the original experiment used an ice-pail.

The amount of opposite, electricity induced on surrounding
conductors hy an electrified body is equal to the hody^s oivn charge.
One body cannot be charged with a quantity of -|-E without
an equal charge of — E being established somewhere else, and
vice versa. The student will bear in mind that whenever in
his experiments he charges any body with electricity, an
equal complementary charge always exists distributed over
neighboring objects or on the walls of the room. When a
thunder-cloud is charged it has its equal complementary
charge in the part of the earth nearest it. The sum total
of all the +E and — E in existence is zero.

408. Definition of electro-static iyiduction. — We are now in
a position to understand the following definition : Electro-
static induction is the action ivhereby a charged body surrounded
by a dielectric evokes an equal and opposite charge on the inner
surface of the enclosure containing the body and the dielectric.

409. Inductive capacity. — The power of transmitting in-
duction varies with different substances. Across glass, sul-
phur, and shellac the effect produced by an electrified body is
different from that across air, the distance being the same.
The power of a dielectric substance to receive and transmit
that electric strain which we call induction depends on the
specific inductive capacity of the substance.

When we electrify a body, a certain quantity of energy is
expended, and this is regarded as. the energy of the electric
charge, and may be recovered by discharging the body. The
energy is, however, stored in the ether around the body said
to be charged.

ELECTP.IC DENSITY.

44T

Section III.

DISTRIBUTION OF ELECTRICITY.

410. Electric density, — The following experiments by no
means give exact quantitative measurements, but they are
suitable for our purpose.

Experiment 1. — a. Charge a well-insulated metal sphere with +E.
Touch some point on the surface with a proof-plane and bring it
in contact with an uncharged electroscope. Notice the amount of
divergence.

6. Discharge the proof-plane and electroscope and touch a different
point on the sphere with the proof-plane, and touch the electroscope as
before. Notice that there is equal divergence of the leaves.

Experiment 2. — a. Electrify an insulated pear-shaped conductor (Fig.

344). Touch the larger end
of the conductor with a proof-
plane and bring it in contact
with an electroscope. Notice
the amount of divergence of
the leaves.

h. Discharge the proof -plane
and electroscope. Touch the
pointed end ; notice that on
bringing the proof-plane in
contact with the electroscope
the divergence of the leaves is
greater than before. In this

Fig. 344.

case, therefore, the charge is not uniformly distributed.

We conclude that the electricity is of eriual density on the
sphere, but of unequal density on the pear-shaped conductor.
Electric density is defined as the quantity of electricity on a
body per unit area.

It follows, from this definition, that if the surface be
increased, the quantity of electricity being the same, the
density is diminished, and vice versa. The relation between

448

ETHER DYNAMICS.

density and area is readily shown in the following manner :
Suspend a sheet of tin foil
from a glass rod (Fig. 345),
connect the lower end of the
foil with an electroscope, and
charge the foil lightly. EoU
the foil up on the rod, and as
the surface becomes reduced
the leaves diverge more widely.
Distribution of electrifica-
tion, or the electric density at

different points on a conductor, depends on its shape. In
Eig. 346 the distances between the surfaces of the bodies and
the dotted lines are intended to represent approximately the
relative densities at different parts of each body.

ilG. 345.

FIG. 346.

411. Effect of points. — As bodies become pointed, the elec:
trie density increases at the pointed end, until it becomes
so great that the electricity is discharged. The particles of
air surrounding the point become heavily charged and are
repelled ; other particles rush in to take their place, and they
in turn are electrified and repelled. A current of air pro-
ceeding from the point, called the " electric wind," is thus
produced, and the conductor becomes discharged by a process

ELECTKOSTATICS AND ELECTKOKINETICS. 449

somewhat analogous to convection of heat. Pomts or sharp
edges on a conductor cause a continuous loss of electricity,
and, therefore, must be carefully avoided in all apparatus
where they are not essential.

Section IV.

ELECTRICAL POTENTIAL.

412. Electrostatics and electrokinetics. — Electricity may be
at rest, as in a charged body, or it may be in motion, as in
the case of a charged body connected by a conductor with the
earth, when it is discharged through the conductor to the earth.
It will be shown later on that as long as a floAv of electricity
continues the conductor along which it flows has properties
different from those of a simple electrified body. That
branch of electrical science which treats of the properties
of simple electrified bodies is called Electrostatics, because
in them electricity is supposed to be at rest y and that branch
which treats of electricity in motion is called Electrokinetics.

413. Eotential. — The fundamental fact of electricity is
that we are able to place bodies in different electrical con-
ditions. A charge of electricity, which implies an abnormal
electrical condition, is the foundation of all electrical phe-
nomena. We are now to discuss in a very simple manner
the meaning and use of the very important term potential
with reference to electricity.

a. When a charged conductor is connected with the earth,
a transfer of electricity takes place between the body and
the earth.

h. If the body be charged with +E, we say arbitrarily that
electricity passes to the earth ; but if the body be charged
with — E, electricity passes /rom the earth to the body.

c. If two insulated charged conductors be connected with

450 ETHER DYNAMICS.

each other, electricity may or may not pass from one to the
other. Now whether electricity passes from one to the other,
and in what direction it passes, if at all, depends upon the
so-Gslled potentials of the conductors.

d. If two bodies have the same potential no transfer of
electricity takes place between them when they are connected
by a conductor ; but if the two bodies have different poten-
tials, there will be a transfer, and the body f7vm which the
electricity flows is said to be at a higher potential than the
one to which it flows.

414. Definition of potential. — The potential of a conductor
may, therefore, be defined provisionally as the electrical con-
dition of that conductor which determines the direction of
the transfer of electricity.

The term potential is relative, i.e. we compare the potential
of one body with that of another.

It is important to have a standard of reference whose po-
tential is considered to be zero, just as it is convenient in
stating the elevations and depressions of the earth's surface
to give the distances above or below sea-level, which is taken
as the zero of hight. For experimental purposes the earth is
usually assumed to be at zero potential. A body charged
with H-E is understood to be one that has a higher potential
than that of the earth, and a body charged with — E is one
that has a lower potential than that of the earth.

415. Analogies. — Potential is analogous, in many respects,
to (1) temperature, and (2) liquid level.

(1) When we say that the temperature of air is 20° or
— 10° C, we mean that its temperature is 20° above or 10°
below the standard temperature of reference, viz. that of
melting ice. If two bodies at different temperatures be
placed in thermal communication, heat will pass from the
body at a higher temperature to the one at a lower, and will
continue to do so until both are at the same temperature.

ELECTRICAL CAPACITY. 451

(2) If two vessels, containing water at different levels, be
put in communication at their bottoms by a pipe, water will
flow from the one at a higher level to the one at a lower until
the Avater is at the same level in both vessels.

Temperature is not heat ; level is not water ; and potential
is not electricity, but merely the state of the conductor which
determines the direction of transfer of electricity.

All points of a conductor^ when the electricity upon it is at rest,
are at the same potential, regardless of any difference of density which
may exist at different points. If it were not so there would be a
continual flow of electricity from the higher to the lower until
equilibrium was established, i.e. until all points had the same po-
tential. We can demonstrate this fact by experiment.

Experiment 1. — Charge a pear-shaped body, A (Fig. 347), with
electricity. Connect a proof-plane with an electroscope and touch
the charged conductor with the
proof-plane at different points ;
the leaves diverge just the same
at all points touched, thus show-
ing that the potential at aU
points is the same, although the
density at different points varies

(compare this experiment with '^"^ yjg. 347.

Exp. 2, Section III).

Observe that it is diffe'rence of potential, or simply potential, and
not quantity or density, which determines a flow of electricity.
Water does not flow from a larger or deeper pond into a smaller
or shallower one unless there is a difference of level.

416. Electrical capacity.

If two conductors of the same shape, and surrounded by the
same dielectric, be charged, it will be found that the larger one re-
quires a larger charge than the smaller one to electrify it to the
same potential ; i.e. the larger one has a greater electrical capacity
than the smaller one. Hence the potential of a conductor depends
upon its charge and its capacity. If C = the capacity of a con-
ductor, Q = the quantity or charge of electricity, and V = the
potential, then ^

452

ETHER DYNAMICS.

rrom this we see that the capacity of a conductor is equal to the
charge necessary to raise its potential from zero to unity.

The capacities of spheres are found to be proportional to their
radii. Thus if a sphere charged with 20 units of electricity, and
having a radius of 4 inches, be brought in contact with an un-
charged sphere having a radius of 1 inch, and these be afterwards
separated, the quantity on the large one will be 16 units ; that on
the small one, 4 units.

Section V.

INDUCTION. ELECTRICAL MACHINES.

417. Electrophorus. — This apparatus is used to produce
electrification by induction. It consists of a shallow iron
dish, A (Fig. 348), filled with sealing-wax. At the center

of the dish is a protuberance, B, which ex-
tends just through the wax. A flat brass
disk, C, has a glass insulating handle.

Experiment 1. — Strike the surface of the wax
a few times with a cat's fur, or rub it with a dry
flannel. The wax becomes electrified with — E.
Place the disk C upon it. The + E of the disk is
hound by the — E of the wax, but the — E of the
disk is repelled by the — E of the wax and passes
through the protuberance B to the dish below,
and thence to the earth. Consequently when the
disk C is raised by the insulating handle from the
wax, it is charged with +E, and the charge can
be transferred to any body {e.g. a Leyden jar, see
§ 420), and then the disk can be recharged by
replacing it on the wax. This may be repeated
many times without sensibly decreasing the charge of — E on the wax.

418. Continuous electropliorus. — Topler-Holtz machine.

Charging by means of the electrophorus like that described above
is necessarily intermittent. The Topler-Holtz machine acts as an
approximately continuous electrophorus, i. e. the act of charging by
this machine is more nearly continuous.

CONTINUOUS ELECTROPHORUS.

453

Fig. 349 represents this machine in perspective, and Eig. 350 is
a diagram of its essential parts. On the back of a stationary glass
plate, F F (Fig. 350), called the field plate, are pasted two quadrants
of varnished paper, 1 1, called the inductors. In front of the field
plate is a revolving glass plate upon which are pasted equidistant
tin-foil carriers, a a a' a' n n\ having a flat metal button on the
center of each. Two brushes of tinsel, C C, connected with the
inductors, are so supported as to touch the buttons as they pass,
and thus a connection is established between the buttons in tem-
porary contact with the brushes and the inductors. A stationary

Fig. 349.

metal rod, A B, has metal combs with pointed teeth attached to it
near each end. The central teeth of these combs are removed and
replaced by tinsel brushes. This rod serves as a conductor between
the two buttons on the same diameter, and may be called the neu-
tralizing conductor. A second pair of combs, C C, are connected
with the separable discharging conductor K K'. Connected with
each part of this conductor is a Leyden jar or condenser.

The mere contact between unlike substances is sufficient to pro-
duce a very small incipient charge which, -as the plate revolves,
rapidly increases to a maximum. In starting this action the two
parts of the discharging conductor are usually brought in contact.
As the two parts, K and K', become oppositely charged, they may be
separated farther and farther apart, and discharges between the two
extremities, in the form of sparks and brushes, occur at intervals.

454

ETHEE DYNAMICS.

which increase with an increase of distance. By the addition of
Ley den jars, which also become oppositely charged, the amount of
charge previous to each discharge is increased, and consequently
the energy of the discharge and the brilliancy of the spark are
increased, though the discharges are less frequent.

As the plate rotates, the two inductors are kept constantly and
oppositely charged, and as two opposite carriers {n and n' for ex-
ample) are about to leave the inductors the following takes place :
At n the positively charged inductor acts through the glass upon the
carrier and comb, attracting and binding the — E and repelling +E.
Similar action takes place at n', but with opposite signs. These

repelled charges unite through the conductor A B and neutralize
each other, leaving the carriers n and n' charged respectively with
— E and +E, As n and n' move away from the inductors their
charges become free, and on reaching the brushes C C they com-
municate a portion of their charges to the brushes to make good
any losses by leakage or otherwise which the inductors may sustain.
We are now able to see how the charges of the inductors are re-
ceived and maintained.

We now turn our attention to the discharging conductor. The
two inductors act inductively upon the two parts K and K' of the
conductor, charging K with +E and K' with — E. The work re-

CONDENSER. 455

quired to keep up the motion of the revolving plate increases as the
charges rise, as there is a constant pulling apart, at different points,
of bodies oppositely charged. Thus mechanical energy becomes
transformed into electric energy, or the energy of ether strain, i

The above is a partial description of the action of this machine.
For a complete description the student may consult larger v^^orks.^

419. Condenser. — A very important adjunct to an electrical
machine is a condenser of some kind, by means of wMcli a large
quantity of electricity can be collected on a small surface.

Experiment 2. — Let a person stand on an insulated stool (§ 400), and
place one hand on the prime conductor of a machine. Let the other
open hand press agamst a plate of glass or disk of vulcanite, held on the
open hand of a second person standing on the floor. After a few turns
of the machine, let the hand that has been on the prime conductor grasp
the free hand of the second person. Quite a shock will be felt by both.
Or the connection may be made through a group of persons having hold
of one another's hands, when the whole company may receive a shock.

It is evident tliat by this process an unusual quantity of
electricity had collected previous to the discharge. This
furnishes an excellent illustration of how electricity may be
bound by inductive action. The explanation is simple. The
hand of the first person, charged with + E, acts by induction
through the glass upon the second person, attracting — E to
the surface of the glass with which his hand is in contact,
and repelling -|- E to the earth. Thus, through their mutual
attraction, the two kinds of electricity become, as it were,
heaped up opposite ?ach other, and yet are prevented, by the
insulating glass, from uniting.

It thus appears that the electrical capacity of a body depends
not only upon its size (§ 410) but upon tJie presence of charges upon
other conductors.

1 After a Topler-Holtz machine has charged a battery of Ley den jars, i.e. stored
up in the jars electrical energy, the belt may be slipped from the machine (to reduce
friction), when the battery will drive the machine, reconverting the energy into
mechanical energy.

2 Consult Barker, Ganot, Gumming, etc.

456

ETHER DYNAMICS.

Fig. 351.

Fig. 352.

Since C = ^ (§ 416), it is evident that the increase in capacity of

the conductor is not due to an increase of potential. An electrical
condenser may be regarded as an appliance for increasing the charge
without increasing the potential.

420. Leyden jar. — One of the most convenient forms of
condenser is the Leyden jar.^ It consists of a glass jar (Fig.
351) coated with tin-foil both inside and outside to about
two-thirds its hight. A brass rod passes inside through a

varnished wooden stopper, and
touches the inner foil, and
terminates in a brass knob on
the outside.

The jar may be charged by
connecting one of its coatings
with the conductor of an elec-
trical machine and the other
with the earth. Or it may be charged by connecting the
outside coating with one of the discharging conductors of
the Holtz machine, and bringing the other pole near to the
ball leading from the inner coating. To discharge the jar,
connect the outer coating with the knob of the jar. To avoid
a shock in so doing, a discharger is used (Fig. 352), which
consists of a bent wire terminating at each end with metal
balls. The wire is held by a glass insulating handle.

421. Capacity of a condenser. — The edacity of a conden-
ser is proportional (1) to the area of the metallic conductor ;

(2) to the specific inductive capacity of the dielectric ; and

(3) is inversely proportional to the thickness of the dielectric.

1 So called because one of the first jars was constructed (by Cuneus) at Leyden,
Holland (1746). The original discovery was, however, made a year earlier by Kleist
of Pomerania. He happened to touch a charged conductor of an electrical machine
with a nail protruding from a bottle containing water. On removing the bottle and
attempting to remove the nail from the bottle he received a violent shock. His
hand on the outside of the bottle and the water on the inside undoubtedly answered
the purpose of coatings.

CONDITION OF THE DIELECTRIC.

457

The low inductive capacity of some kinds of glass renders it
entirely unsuitable for this purpose.

To secure greater capacity than a single jar of ordinary
capacity will afford, several jars, constituting a " battery " of
jars (Fig. 353), are placed upon a sheet of tin-foil so as to
connect all the outer coatings, while the inner coatings are

Fig. 353.

connected by a wire joining their projecting rods. The sev-
eral jars are by this means practically converted into one
large jar..

422. Conditmi of the dielectric. Seat of charge. — That
inductive action is attributable to the dielectric, and not to
the conductor, is shown by the Leyden jar with movable
coatings (A, Eig. 354). B is the dielectric ; C is the outer and
D the inner conductor. The several parts being put together,
the jar is charged in the usual manner and placed upon an
insulator. Then the inner conductor, D, is raised by a glass
rod out of the jar, and afterwards the glass vessel, B, is
removed from the outer coating. The several parts are now
tested with an electroscope. The coatings produce little or
no disturbance of the leaves ; the glass causes a divergence of

458

ETHEH DYNAMICS.

the leaves. On putting the parts together again and dis-
charging in the usual way, there will be nearly as brilliant a
spark as if the charged jar had not been dissected.

This experiment demonstrates that (1) the seat of the
charges is on the surface of the glass and not on the coat-
ings ; (2) the coatings serve merely the purpose of conductors
to spread electricity at the time of charging, and to allow its
escape from all parts of the electrified surfaces at the time

Fig. 354.

of discharge ; (3) a charge is not an electrification of the
conductors, but, rather, of the dielectric, or, as we shall say
later on, of the 'Afield" itself, the extent of the conductor
determining the limits of the field.

423. Limit of the charge of condensers. — There is a limit
beyond which a condenser cannot be charged. When elec-
trification takes place, the stress produced by the opposite
charges causes a strain in the glass. When the strain be-
comes too great, a discharge occurs across the dielectric, either
through the air over the top of the jar, or, if the glass be thin
enough, by puncturing it.

CONTACT ACTION.

459

Section VI.

ELECTROSTATIC LINES OF FORCE. FIELD OF FORCE.

424. Contact action.

Bring two bodies of dissimilar nature {e.g. a §tick of sealing-wax
and a woolen cloth) in contact, best by rubbing to secure better
contact, and separate them, and they exhibit strong attraction for
each other, ordinarily vastly greater than that of gravitation. The
contact serves to establish bonds of attraction, i.e. the ether between
them is supposed to operate like india-rubber bands, pulling the two
bodies together. The bodies are thus said to be electrically excited.
To separate the excited bodies requires work to be done ; and the
bodies when separated possess energy of electrical separation.

425. Lines of force. Field of force.

The space or dielectric between and, to a limited extent, around

the excited bodies is assumed by Faraday to be full of what he

called lines of force., the positive direction of a line at any point in

this space being the direction in which a positively electrified particle

tends to move under the influence of the electrical field. The space

thus occupied by lines of force is called the field of force. Faraday
remarks that the stress is as if these lines were stretched
elastic threads endowecf with the property of shortening
themselves and also the property of repelling one another
as well. In other words, there is a tension along these lines
and a pressure at right angles to them. When bodies
oppositely excited are brought near together, the lines are

almost straight from one to the other (except near the edges) of the

facing areas, as shown in

Fig. 355. As they are

more separated the lines

curve outward, always

tending to separate from

one another and from the

common axis of the two

bodies, some even curling

round to the back of the

bodies, as represented in

Fig. 356. The expression

" lines of force " must be Fig. 356.

^=^

Fig. 355,

460 ETHER DYNAMICS.

regarded as purely a matter of convenience. They have no more
and no less existence than have "rays of light."

Section YII.
atmospheric electricity.

426. Lightning. — Franklin, by a series of historic experi-
ments, proved the exact similarity of lightning and thunder
to the light and crackling of the electric spark. Certain
clouds which have formed very rapidly are highly charged,
usually with +E, but sometimes with — E. The, surface of
the earth and objects thereon immediately beneath the cloud
are, of course, charged inductively with the opposite kind of
electricity. The cloud and the earth correspond to the coat-
ings, and the intervening air to the dielectric, of an immense
condenser. The opposite charges on the earth and on the
cloud hold each other prisoners by their mutual attraction.

As condensation progresses in the cloud its capacity de-
creases and its potential rise's (since CV^Q). This process
continues till the difference of potential between the cloud
and the earth becomes great enough to produce a discharge
through the air.

It is the accumulation of induced charges on elevated
objects, such as buildings, trees, etc., that offers an intensi-
fied attraction for the opposite electricity of the cloud in
consequence of their greater proximity, and renders them
especially liable to be struck by lightning.

The clouds gather electricity from the atmosphere. Our
knowledge of the method by which the atmosphere becomes
charged is very limited.

427. Lightning-rods. — A good lightning conductor offers a
peaceful means of communication between the earth and a
cloud ; it leads the electricity of the earth gently up toward

THE AURORA. 461

the cloud, and allows it to combine with its opposite without
disturbance, thereby so far discharging the cloud as possibly
to prevent a lightning stroke ; or, if the stress be too great to
be thus quietly disposed of, the flash strikes downward, and
is led harmlessly to the earth by the conductor. A71 ill-con-
structed lightning-rod may be luorse than 7ione. A good rod
should be made of good conducting material, so large that it
will not be melted, and free from loose joints. The lower
end should be buried in earth that is always moist, and the
upper end should terminate in several sharp points. Maxwell
suggests that the best form for a lightning protector is one
which approximates a net-work covering the entire house
(see foot-note, p. 445).

428. The aurora is a luminous phenomenon caused (as
experiments performed by Lemstrom in Lapland seem to
indicate) by currents of electricity passing from the higher
and rarefied regions of the atmosphere to the earth. In the
Arctic regions the aurora borealis (northern lights) is of
almost daily occurrence. It sometimes forms an arch, and
sometimes illuminates the whole sky.

462 ETHEE, DYNAMICS.

CHAPTEE III.

ENEEGY OF ELECTRIC ELOW. ELECTRO-KINETICS.

Section I.

INTRODUCTORY EXPERIMENTS.

429. Apparatus required.

There are required a condensing electroscope, i.e. one whicli has two
disks separated from each other by a dielectric (Fig. 358), the upper one
having an insulating handle ; a tumbler | full of water, into which have
been poured two or three tablespoonfuls of strong sulphuric acid ; a strip
of sheet-copper, and two pieces of rolled zinc, each about 5 inches long,
1^ inches wide, and at least ^-^ of an inch thick (a piece of No. 16 copper
wire 12 inches long should be soldered to one end of each piece of metal,
and the soldering covered with asphaltum paint) ; 2 yds. of silk insulated

No. 18 copper wire; two double
connectors (Fig. 357) which serve
to join two wires without the in-
convenience of twisting them to-
FiG. 357. gether ; and a battery of four vol-

taic cells (either Bunsen or other
reputable kind). One of the zincs should be amalgamated as follows :
First dip the zinc, with the exception of i inch at the soldered end, into
the acidulated water ; then pour mercury over the wet surface, and
finally rub the surface, now wet with mercury, with a cloth (to insure
complete amalgamation, it is best to repeat this operation).

430. Experiments.

Experiment 1. — a. Put the unamalgamated zinc into the tumbler
containing acidulated water. Bubbles of hydrogen gas arise from the
surface of the immersed zinc.

6. Remove this zinc and introduce the amalgamated zinc. No bubbles
(or at least very few) arise from the latter, provided that the zinc is
properly amalgamated.

LESSON LEARNED.

463

c. Put the copper strip into the liquid, hut do not allow the two metals
or their wires to touch. No bubbles arise from either metal. Connect
the wires of the two metals with a double connector ; copious bubbles
arise from the copper strip, but very few from the zinc strip. Bubbles
escaping from the copper make it appear as if chemical action were taking
place between the metal and the liquid. But experience will teach you
that the appearance is deceptive, as you will find that in no case is
copper consumed.

d. Substitute the unamalgamated zinc for the amalgamated ; bubbles
rise abundantly from the surfaces of both the zinc and copper.

Lesson learned : An unanialgamated zinc is acted on by the
liquid under all circumstances ; an amalgamated zinc is not
acted on by the liquid unless the copper strip is also in the
liquid, and not then unless the metals are connected. If then
we would at any time stop the action, we have only to dis-
connect the metals. It seems also that the wire connecting
the two metals serves some important purpose in keeping up
this action.

Experiment 2. — In this experiment it will be necessary to use metal
plates of much larger size, or (which will prove much more satisfactory)
we must use an apparatus somewhat in advance of our present knowledge,
mz. a battery (§ 4S1) of (say) four
cells connected in series (§ 482).

a. Connect the copper to the
lower disk of. the electroscope (Eig.
358) by an insulated wire, merely
touching it with the end of the
wire, and the zinc to the upper
disk. Remove the wires, and lift
the top disk by the insulating han-
dle. Tlie leaves of the electroscope
diverge. Prove by suitable test that
the electrification m the leaves is
positive.

h. Repeat this operation, but
touch the lower disk with the wire from the zinc, and the upper one with
the wire from the copper. Show that the leaves have now a negative
charge.

Fig. 358.

464 ETHER DYNAMICS.

When the upper plate is lifted, the capacity of the condenser
diminishes considerably, so that the small charge on the lower disk
raises its potential so much that the gold leaves diverge. This will

be understood from the equation V = — (§ 416); when the denomi-

nator C of the fraction — diminishes, the fraction increases, and
therefore V increases.

If a plate of metal be placed in a liquid of a class which we
shall term an electrolyte (i.e. one which is capable of being
decomposed by a current of electricity), there is a difference
of electrical condition produced between them so that the
metal becomes either of higher or lower potential than the
liquid, according to the nature of the metal and liquid.

W-e know that if two conductors be at different potentials,
electricity tends to flow from the one whose potential is
higher to that whose potential is lower ; if, therefore, two
dissimilar metals be placed in the same electrolytic liquid
and we show by actual experiment, as above, that the free
end of the wire in connection with one plate is charged with
+ E, and the free end of the other with — E, we conclude that
if the two oppositely charged bodies be brought in contact, a
current of electricity will flow from the positively charged
plate to the negatively charged one. A current therefore
flows through the connecting wire from the copper (which is
called the positive electrode) to the wire leading from the zinc
(which is called the negative electrode), when they are con-
nected.

That difference in quality, in virtue of which zinc and cop-
per placed in acidulated water can give rise to an electric
current, is called their electro-chemical difference, and the zinc
is said to be electro-positive to the copper in the liquid.
There is a perplexing nomenclature in use by which the zinc
plate is called the electro-positive element or plate, though it
is called the negative pole of the combination.

VOLTAIC CELL. 465

Section II.

VOLTAIC BATTERIES. ELECTRIC CIRCUITS.

431. Voltaic cell. — Two electro-clieniically different solids
(of which zinc is almost invariably one) placed in an electrolytic
liquid constitute what is called a galvanic or voltaic ^ cell (or
2)air). One of these plates must be more actively attacked
by the liquid than the other ; the plate most acted upon is
called the electro-positive plate, and the other the electro-
negative one.

The greater the disparity hetiveen the two solid elements ivith
reference to the action of the liquid on them, the greater the
difference in potential ; hence, the greater the current.

In the following electro-chemical series the substances are so
arranged that the most electro-positive, or those most affected
by dilute sulphuric acid, are at the beginning, while those
most electro-negative, or those least affected by the acid, are
at the end. Tlie arrow indicates the direction of the current
through the liquid.

0^ ^ ri o
.:1 2 .S <X) O r;^ ^ eg

It will be seen that zinc and platinum are the two sub-
stances best adapted to give a strong current.

When the wires from the two plates are joined the dis-
charge of the two plates would produce electrical equilibrium,
were there not some means of maintaining a difference of
potential between the two plates. This is accomplished by

1 The original Volta's cell consisted of a plate of copper and a plate of zinc im-
mersed in dilute sulphuric acid. It is a very inefficient battery, yet with such an
appliance Sir Humphry Davy performed his classic experiments upon the metals of
the alkalis, and produced the first voltaic arc.

466 ETHER DYNAMICS.

the chemical action between the liquid and the electro-positive
plate and at the expense of the chemical potential energy of
the electrolyte and plate. A voltaic cell is, therefore, a con-
trivance which converts chemical energy into electrical energy}
It should be remembered that it is the role of the battery to
maintain a difference of potential between the two plates, or
what is the same thing, between the battery terminals or
poles, and not to '• generate electricity."

432. Circuit. — This term is applied to the entire path
along which electricity flows, and it comprises the battery
itself and the wire or other conductor connecting the bat-
tery-plates. Bringing the two extremities of the wire in
contact and separating them are called, respectively, closing
and opening, or making and breaking, the circuit. Opening a
circuit at any point and filling in the gap with an instrument
of any kind so that the current is obliged to traverse it, is
called introducing the instrument into the circuit.

433. Ground circuit, — It was an early discovery in tele-
graphic history that a complete metallic circuit is not neces-
sary, but that, in common parlance, the earth can be used as
a "return circuit." This type of circuit is represented by
a battery with a wire leading from one plate to any con-
venient point of the earth, and a second wire leading from
the other plate to any other point of the earth, which may be
many miles distant from the first point. No one can assert
that the current in such a case really goes through the earth
from one of these points to the other. The earth may be
regarded as a great resei^voir, rather than as a conductor.
The battery acts like a pump raising electricity at one end
from the earth and discharging it at the other end to the
earth. It is obvious that a pump might be kept in action by
pumping from an ocean and back into the same ocean with-
out disturbing its level and yet there would be a continuous

1 A single voltaic couple is lasually termed a cell ; a combination of cells, a battery.

ELECTRO-CHEMICAL ACTION OF THE BATTERY. 467

flow through the water conductors or pipes. This would
represent what is known as a ground circuit.

434, Theory of the electro-chemical action of the battery. —
The following is a brief statement of Grothus' theory of
electrolytic action, somewhat modified. The small ovals in
Figs. 359 and 360 represent molecules of the electrolyte,
which in this case we suppose to be water,^ each molecule of
water containing two atoms of hydrogen (Hg) and one atom

Fig. 359.

Fig. 360.

of oxygen (0). The series of molecules a illustrates the
condition of the molecules before the metals are introduced ;
the series h represents their condition after the introduction
of the metals and before the circuit is closed. Eig. 360 rep-
resents the condition of things at the instant the circuit is
closed or at the instant of discharge. The molecules are
polarized and like sides turned in the same direction. At
this instant there is a redistribution of electricity, an electric
flow, an equalization of potential. At this instant, chemi-
cally speaking, there is as it were an interchange of partners,
the next the zinc combining with it to form ZnO, its Hg

1 It is, perhaps an open question whether the water (HgO) or the acid (H2SO4) is
the real electrolyte. For our purpose it does not matter.

468 ETHER DYNAMICS.

combining with the of the next molecule, and so on through
the whole row of molecules, until, finally, the Hg appears at
the surface of the copper, where it is set free. Immediately
following the discharge and equalization of potential a re-
polarization (p) takes place followed by a discharge (c). The
discharges follow one another so rapidly as to furnish prac-
tically a continuous flow of electricity.

435. Electrolytic conduction. - — As rapidly as ZnO (zinc
oxide) is formed it combines with the sulphuric acid and
forms zinc sulphate. The hydrogen escapes in bubbles, as
has been shown, from the electro - negative plate. Thus
oxygen keeps disappearing by combination with the zinc at
one end of the electrolyte ; the hydrogen, by evolution as a
gas, disappears at the other. Meantime the molecules keep
exchanging atoms, so that a constant traveling of the H and
atoms through the electrolyte is kept up, each conveying
its own peculiar charge. In this way what is virtually a
current traverses the liquid. The current, however, does not
traverse the liquid as it does a wire, but electricity is trans-
mitted by electrolytic action. The electricity flows not
through but with the atoms of matter. It is conveyed
through the liquid by a procession of charged atoms, and the
process of transmission is more nearly that of convection
than of conduction. This process is called electrolytic coov-
ductio7i.

Section III. .

SOME DEFECTS OF BATTERIES.

436. Importance of amalgamating the zinc. — All commer-
cial zinc contains impurities, such as carbon, iron, etc.
Fig. 361 represents a zinc element having on its surface a
particle of carbon a, purposely magnified. If such a plate be
immersed in dilute sulphuric acid, the particles of carbon will

POLARIZATION OF THE NEGATIVE ELEMENT. 469

form with the zinc numerous voltaic circuits, and a transfer
of electricity along the surface will take place. This coast-
ing trade, as it were, between the zinc and the impurities on
its surface, diverts so much from the regular bat-
tery current, and thereby weakens it. In addi-
tion to this, it occasions a great waste of materials,
because, when the regular circuit is broken, this
local action, as it is called, still continues. If
pure zinc^ were used, no local action would occur
at any time, and there would be no consumption of
material except when the circuit was closed. If
mercury be rubbed over the surface of the zinc
after the latter has been dipped into acid to clean
its surface, the mercury dissolves a portion of the zinc, forming
with it a semi-liquid amalgam, which covers up its impurities,
and the amalgamated zinc then comports itself like pure zinc.
437. Polarization of the negative element.

Experiment. — Construct a voltaic cell composed of dilute sulphuric
acid and plates of copper and zinc. Introduce into the circuit a gal-
vanoscope (§ 447) and note the deflection of the needle when the circuit
is first closed. Watch the needle for a time. Little by little this deflec-
tion will decrease, and as it decreases bubbles of gas collect on the copper
plate. This accumulation of gas is called " polarization ^ of the negative
element or plate."

We already understand that a difference of potential is the
indispensable prerequisite to a flow of electricity. Accom-
panying a difference of potential there seems to be something
analogous to a force which is said to cause the flow of, or to
urge, the electricity through the circuit. The film of gas on
the copper reduces the electro-chemical difference between it
and the zinc plate, upon which the generation of this force
depends, and thereby diminishes the efficiency of the battery.

1 rormerly pure zinc, obtained at great expense by distillation, was used.

2 The term polarization in common use is a most senseless term as here applied.
Polarization in the electrical world is made to cover a multitude of sins.

470

ETHER DYNAMICS.

Instead of a copper-zinc pair, we soon have a hydrogen-zinc
pair. A single fluid battery cannot, therefore, yield a current
of constant strength unless some means is used to remove the
hydrogen.

To overcome this defect some arrangement must be adopted
to prevent this deposit of hydrogen upon the negative ele-
ment. The usual method is to employ in addition to the
dilute sulphuric acid, which we will term the exciting liquid,
some other substance (usually a liquid) which is a strong
oxidizing agent, i.e. which can combine with and remove the
hydrogen as soon as it . is liberated at the negative plate (or
positive pole). A substance used for this purpose is termed
a depolarizer. A mixture of a solution of crystals of bi-
chromate of potassium in water with a suitable quantity of
dilute sulphuric acid, forms a depolarizer such as is used in
the so-called hichromate batteries. Other depolarizing sub-
stances in common use are bi-
chromate of sodium, nitric acid
(an excellent depolarizer but
very objectionable on account
of the corrosive and unwhole-
some fumes to which it gives
rise), chromic acid, peroxide of
manganese, copper sulphate,
etc.

438. Grenet cell.

This is a bichromate of potassium
battery in which two carbon plates,
C C (Fig. 362), electrically connected,
and a zinc plate, Z, suspended be-
tween them by a brass rod, a, are
immersed in the mixed liquid referred to above.

This combination furnishes a much more energetic and constant
current than would be furnished if only dilute sulphuric acid were
used. But although polarization of the negative element is dimin-

FlG. 362.

BUNSEN CELL.

471

ished, another detrimental action is substituted. The layer of
solution next to the carbon plate is soon (say after a constant use
of half an hour to an hour) deprived of its active oxidizing powers,
and then polarization of the negative plate and consequent w^eak-
ening of the current set in. This difficulty may be partially
remedied by occasionally agitating the liquid or allowing it to rest,
during which time the various portions of it become homogeneous
by diffusion, and the liquid near the carbons becomes more active.

439. Biinsen cell.

A plan generally adopted to keep the oxidizing liquid away from
the zinc plate, where it is not wanted and only does harm, is to
place the carbon plate in an unglazed, porous, earthen cup and to

Fig. 363.

Fig. 364.

surround it with the oxidizing substance. This arrangement, called
a two-fluid cell, is that adopted by Bunsen (Fig. 363), Grove, Fuller,
and others.

In any form of two-fluid cell yet devised, the oxidizing fluid
sooner or later diffuses through the porous cup and reaches the
zinc. Destructive action on the zinc then begins, which amalga-
mation cannot prevent, and a portion of zinc is uselessly consumed
without effecting anything in the way of generation of electric cur-
rent. An attempt to prevent this trouble is sometimes made by
using a solid depolarizer.

472

ETHER DYNAMICS.

440. Leclanche cell.

There is a class of galvanic cells in which the negative element
is protected from polarization by means of metallic oxides. Of
these the best known is the Leclanche cell (Fig. 364). In this cell
the carbon plate C is contained in a porous cup P, and packed
round with fragments of gas-retort coke and manganese peroxide.
The manganese compound has a strong afiSnity for the hydrogen.
But the chemical action of solids is sluggish and they quickly
polarize when in action. They need periodical rest to recover their
normal condition. Such are called open-circuit batteries, since they
are suited for work only on lines kept open or disconnected most
of the time, such as in telephone and bell-ringing circuits. The
zinc element, Z, which is a rod of zinc, is immersed in a solution
of ammonium chloride, which is the exciting liquid.

441. Daniell cell.

Leaving the hydrogen-generating batteries, we will examine
briefly another form incapable of this species of polarization. The
Daniell cell (Fig. 365) uses a solution which, instead of depositing
hydrogen, deposits copper upon a copper
negative plate, and hence is free from hy-
drogen polarization. It contains a copper
negative and a zinc positive plate. The
copper plate is immersed in a solution of
copper sulphate, the zinc in a solution of
zinc sulphate or dilute sulphuric acid, and
a porous cup separates the two liquids. By
the electrolytic action, the zinc combines
with the sulphuric acid (H2SO4) forming zinc
sulphate (ZnS04), thereby setting hydrogen
free. This hydrogen, while on its way to
the negative element of the copper plate,
meets the copper sulphate solution (CUSO4)
which it decomposes, forming sulphuric acid
again (H2SO4), and setting free the copper,
which is deposited on the copper plate.

In some copper sulphate batteries the porous cup is not employed,
the difference in the specific gravity of the solutions being relied
upon to keep them separate, as in the so-called gravity cell (Fig.

Fig. 365.

DANIELL CELL.

473

366). This form of cell is commonly used on closed circuits
The chief merits of these cells are
the complete absence of polari-
zation of the negative plate (con-
sequently, the constancy in the
potential difference of the two
elements) and the uniformity of
current which is yielded.

The kinds of cells that have
been devised are numberless. The
voltaic battery will probably long
continue to be of scientific inter-
est, but for commercial uses it
has already become well-nigh ob-
solete, being replaced by other
cheaper and more efficient ap-
pliances for generating electric
currents, to be hereafter described.
The battery must, therefore, com- Fig. 366.

mand in the future relatively less attention in our text-books
than formerly.

Questions.

1. a. What is an amalgamated zinc plate ? 6. A voltaic cell or pair?
c. An electrode ?

2. a. How may it be shown that a plate of copper and a plate of zinc
become electrified when placed in dilute sulphuric acid ? b. Which will
be positively electrified ? c. If wires leading from these plates be joined,
what change will occur ? d. What purpose does the connecting wire
serve ? e. Why ought not the plates of a voltaic cell to touch each other ?
/. Why ought not the wires of certain voltaic cells to touch each other
when not in use ?

3. a. What do you understand by an electric current ? b. What is
the function of a battery ? c. With what propriety is the zinc plate of. a
voltaic cell called the positive plate and the negative electrode of a voltaic
system ?

4. How does electricity pass from plate to plate within a voltaic cell ?

5. a. What is meant by local currents ? 6. How may they be pre-
vented ?

474 ETHER DYNAMICS.

6. a. Give an example of a counter electromotive force. 6. By what
other name is it generally known ? c. What harm does it do ? d. How
may it be prevented ?

7. If you have the two leading wires of a battery, and the battery be
concealed, how can you tell which of the wires is connected with the zinc
plate of the battery ?

Section TV.

EFFECTS PRODUCED BY THE CURRENT.

442. Summary of effects. — The several effects producible
by an electric current are as follows : —

(1) Certain compounds in solution can be decomposed by
causing the current to pass through the solution. This opera-
tion is called electrolysis.

(2) A magnetic needle suspended on a vertical pivot will
be deflected if a wire through which a current is flowing be
brought near and parallel to it.

(3) If a wire carrying a current be wound round a rod of
soft iron, the iron becomes a temporary magnet ; the magnet
is called an electro-magnet.

(4) Heat is generated in a wire through which a current
flows.

(5) If the temperature be raised sufficiently at any point
of the circuit, the conductor at that point becomes luminous,
as, for example, in the production of the electric light.

(6) Various physiological effects are produced by a current,
such as " shocks," a peculiar taste when the poles are applied
to the tongue, the sensation of light, etc.

The effects may be classified as electrolytic, magnetic, in-
cluding (2) and (3) ; heating, including (4) and (5) ; and
'physiological.

ILLUSTRATIVE EXPERIMENTS.

475

443. Illust7'atwe experiments. (1) Electrolysis.

Experiment 1. — Take a dilute solution of sulphuric acid (1 part by
volume to 20), pour some of it into the funnel (Fig. 367), so as to fill the
U-shaped tube when the stoppers are re-
moved. Place the stoppers which support
the platinum electrodes tightly in the tubes.
Connect with these electrodes the battery
wires. Instantly bubbles of gas arise from
both electrodes, accumulating in the upper
part of the tube and forcing the liquid back
into the tunnel. Close the passage in the
rubber tube by turning down the screw of
the pinch-cock a. Light a splinter of fine
wood, blow out the flame, leaving it glow-
ing ; remove the stopper holding the +
electrode and introduce the glowing splinter
into the gas in this arm of the tube. It
relights and burns vigorously, showing that
the gas is oxygen. Platinum electrodes are
used, otherwise a portion of the oxygen
carried to the + electrode would not be set
free, but would oxidize the metal {e.g.
copper), instead of appearing as a gas in
this arm of the tube. Fill this arm of the
tube with water and stopper it. Invert the
U-tube ; the gas in the other arm of the
U-tube collects in the bend of the tube and
in the small branch tube. Light a match, remove the rubber tube, and
quickly bring the match near the orifice of the branch tube. The gas
burns ; it is hydrogen.

If the experiment be performed with a Hoffman's voltameter'^
(Fig. 368), the gas given off at each electrode may be meas-
ured by the graduations on the arms of the tube. It will
be found that the volume of hydrogen given off is just double
that of the oxygen liberated in the same time. There is another
important use to which this apparatus is put worth men-

FlG. 367.

1 Any vessel employed for performing and measuring electrolysis is called a
voltameter.

476

ETHER DYNAMICS.

BATTERY

Fig. 368.

tioning at this point, though a little out of place. The
quantity of gas given off divided by the
number of seconds the current has been
flowing is a direct measure of the mean
strength of the current, i.e. the number
of units of electricity which flow through
the circuit in one second. Hence this
apparatus serves to measure current
strength.

The electrode by which the current en-
ters the electrolyte is called the anode;
and that by which the current leaves,
the cathode. The elements into which the
electrolyte is broken up are called the ions.
The ion appearing at the anode is the
anion; and that at the cathode, the cation.
444. Relation of the electrodes and battery elements. — It
will be observed that in both the battery and the voltameter
the hydrogen is lib-
erated at the plate
toward which the
current is flowing,
as shown in the
diagram (Fig. 369).
Hydrogen and all
metals appear on
the plate toward which the current flows, whether in the
decomposing cell or the battery ; for example, the silver in
Exp. 4 appears at the cathode, and the copper in the Daniell
cell (§ 441) is deposited on the electro-negative plate.

Experiment 2. — Take in a test-tube a quantity of an infusion of purple
cabbage prepared by steeping its leaves until well cooked. Pour into
this infusion a few drops of any alkali, such as a solution of caustic soda.
The infusion is changed thereby from a purple to a green color. In another

I , , I VOLTAMETER

Fig. 369.

ILLUSTRATIVE EXPERIMENTS.

477

Into this pour a
The purple is

test-tube take another portion of the purple infusion.
few drops of any acid, such as dilute sulphuric acid
changed to a red. Only acids will turn this infusion
to a red and only alkalies will turn it to a green.
Into a rather strong solution of sodium sulphate
pour enough of the purple infusion to give it a
decided color.

Pour some of this colored solution into a V-shaped
glass tube (Fig. 370). Into each arm of the tube in-
troduce a platinum electrode and join these to the
battery wires. Soon the liquid around the cathode
is turned green, while that around the anode is turned
red. Evidently, decomposition of the sodium sulphate has taken place.
An acid and an alkali are the results.

When a chemical salt is electrolyzed, the base appears at the
cathode, and the acid at the anode.

Experiment 3. — Dissolve, by heating, about three grams of pulverized
potassium iodide in about a tablespoonful of water. Make a paste by
boiling pulverized starch in water. Take a portion of this paste about

Fig. 370.

Fig. 371.

the size of a pea, and stir it into the solution.

Wet a piece of writing paper with the liquid

thus prepared. Spread the wet paper smoothly fig_ 372.

on a piece of tin, e.g. on the bottom of a tin

basin (Fig. 371). Press the negative electrode of the battery against an

uncovered part of the tin. Draw the positive electrode over the paper.

A mark is produced upon the paper as if the electrode were wet with a

purple ink. In this case the potassium iodide is decomposed, and the

iodide combining with the starch forms a purplish blue compound.

Experiment 4. — Dissolve about 3 g. of silver nitrate in 100 cc of
water. With this solution nearly fill the electrolysis tank (Fig. 372)

478

ETHER DYNAMICS.

which accompanies the porte-himiere (§ 323). Place the tank in the
porte-lumi^re in position to be projected upon a screen in a dark room.

Connect tlie battery wires with the elec-
trodes in the tank. A beautiful deposit
of silver will be made on the cathode,
spreading therefrom toward the anode,
and bearing a strong resemblance to vege-
table growth ; hence it is called the ' ' sil-
ver tree." In Fig. 373, A represents a
silver tree deposited from a weak solu-
tion, and B one from an extremely weak
solution.

445. (2) Magnetic action and
magnetic field of a straight cur-
rent. Magnetic lines of force.

Experiment 5. — Construct a low re-
sistance battery of (say) four cells. Close
the circuit and dip the wire into a little
heap of filings of soft iron. On raising
the wire you will iind filmgs adhering
374).

If a wire bearing a very strong current be passed vertically through
the center of a board on which have been sifted some very fine iron

Fig. 373.
in a cluster to the wire (Fig.

filings, the filings will arrange them-
selves in circular lines round the
current-carrying wire (Fig. 375), thus
furnishing a graphic representation
of the magnetic field set up by a cur-
rent. If a small pocket compass be
carried around and near the wire, the
needle will at every point take a position tangent to these circular lines

ILLUSTRATIVE EXPERIMENTS.

479

of filings, whichever way the current passes. If the current be reversed,
however, the position of the n and s poles of the needle will be reversed.
This clearly indicates that there is a difference of direction of these
circular lines according as the current flows in one direction or in the
other. These circular lines represent the so-called magnetic lines of force
which occupy a limited space or field round a current-bearing wire.

446. Deflection of the magnetic needle by a current.

Experiment 6. — a. Place the apparatus (Fig. 376) so that the mag-
netic needle, which points (nearly) north and south, shall be parallel with
the wires Wi and W2. Introduce the + electrode of a battery into
screw-cup T2, and the — electrode into screw-cup Ti, and pass a current

Fig. 376.

through the upper wire. At the instant the circuit is closed the needle
swings on its axis, and after a few oscillations comes to rest in a position
which forms an angle with the wire bearing the current.

b. Break the circuit by removing one of the wires from the screw-cup.
The needle, under the influence of the magnetic action of the earth,
returns to its original position.

c. Eeverse the current by inserting the + electrode of the battery into
screw-cup Ti, and the — electrode into screw-cup Tg. Again there is a
deflection of the needle, but the direction of the deflection is reversed ;
that is, the north-pointing pole (N-pole), which before turned to the
west, is now deflected toward the east.

d. Place your right hand above the wire with the palm towards the
wire, and with the fingers pointing in the same direction as that in which

480

ETHER DYNAMICS.

the current is flowing, and extend your thumb at right angles to the
direction of the current (Fig. 377). You observe that your thumb points
in the same direction as the N-pole of the needle under the current-bearing
wire.

e. Eeverse the current again (so that it will flow northward), place
your right hand as before (viz. with the palm towards the wire, and with
the fingers pointing in the same direction as the current) ; your out-
stretched thumb still points in the same direction as the N-pole of the
needle.

/. Introduce the + electrode of the battery into screw-cup T3 and the
— electrode into screw-cup T4 so that the current will flow northward
under the needle. Place the right hand as directed before, except that
it must be under the wire, so that the wire shall be between the hand

OF CURRENT

Fig. 377. — Kight hand above the wire ;
needle below it.

Fig. 378. —Bight hand below the wire ;
needle above it.

and the needle ; the thumb will point in the same direction as the N-pole
(Fig. 378). Eeverse the direction of the current in this wire, and apply
the same test ; the same rule holds.

The rule for determining the direction of the deflection of
the IST-poles of a needle when the direction of the current is
known is this : Place the outsti^etched right hand over or under
the wire so that the luire shall be 'hetiveen the hand and the
needle, with the palm towards the needle, the fingers pointing in
the direction of the current and the thumb extended laterally at
right angles to the direction of the current; then the extended
thumb will point in the direction of the deflection of the N-pole.

Conversely, the direction of the current may be determined
by ascertaining the direction of the deflection it produces, as
follows : Place the outstretched right hand over or under the

ILLUSTRATIVE EXPERIMENTS. 481

wire (always allowing the wire to come between the hand and
the needle) ivith the palm toivard the needle and the extended
thumb in the same direction as the N-pole of the needle is
deflected; then the fingers luill point in the direction the
curre7it is floiving.

It will be observed that a deflection is reversed either by
reversing the current or by changing the relative positions of
the wire and needle, e.g. by carrying the needle from above
the wire to a position below it.

The force exerted by the current upon the needle in deflect-
ing it is called an electro-magnetic force.

447. Simple galvanoscope or current detector.

Experiment 7. — Introduce the + electrode of the battery into screw-
cup T2 (Fig. 376) and the — electrode into screw-cup T3, so that the
current will pass above the wire in one direction and below it in the
opposite direction, as indicated by the arrows. A larger deflection is
obtained than when the current passes the needle only once.

If the right-hand test be applied it will be seen that the
tendency of the current, both when passing the needle in one
direction above and in the opposite direction below, is to
produce a deflection in the same direction, and consequently
the two parts of the current assist each other in producing a
greater deflection.

If a more sensitive instrument, i.e. one which will produce
considerable deflections with weak currents, be required, then
it will be necessary to pass the current through an insulated
wire wound many times around the needle, as shown in the
sectional elevation and plan (Fig. 379). Such an instrument
is called a galvanoscope or current detector, since one of its
important uses is to detect the presence of a current. A
graduated card divided off like that of the mariner's compass
is placed beneath the needle so that the number of degrees
of deflection may be read from it. Fig. 380 represents a

482

ETHER DYNAMICS.

portable detector used by telegraph and telephone line-men
to detect whether a circuit is complete, to locate faults, etc.
From the number of degrees of deflection an approximate

T+ " GALVANOMETER COIL

Fig. 380.

idea of the strength of the current is obtained. The magnetic
needle is inside the box, that outside being merely an indicator
attached to the same pivot.

448. (3) Mag7ietizing effect of an electric current. Electro-
magnets.

Experiment 8. — a. Wind an insulated copper wire in the form of a
spiral round a rod of soft iron (Fig. 381). Pass a current of electricity
through the spiral, and hold an iron nail near the end of the rod.
Observe, from its attraction for the nail, that the rod is magnetized.
A magnet may be provisionally defined as a body which attracts iron.

h. Break the circuit ; the rod loses its magnetism and the nail drops.

The iron rod is called a core, the coil of wire a helix, and
both together are called an electro-magnet. In order to take
advantage of the attraction of both ends or poles of the
magnet, the rod is most frequently bent into a U-shape
(A, Fig. 382). More frequently two iron rods are used,

ILLUSTRATIVE EXPERIMENTS. 483

connected by a rectangular piece of iron, as a in B of Fig.

382. The method of winding is such that if the iron core of

the U-magnet were straightened, or the two

spools were placed together end to end, one

would appear as a continuation of the other.

A piece of soft iron, h, placed across the ends

and attracted by them is called an armature.

The piece of iron a is called a yoke.

449. (4) (5) Heating and luminous effects

of the electric current. — Construct a low re-
sistance battery (§ 481) of four to six cells,

and introduce into the circuit a platinum

wire, No. 30, about ^ inch long. The wire

very quickly becomes white hot, i.e. it emits

white light, which indicates a temperature

of approximately 1900° C.

This experiment illustrates the conversion of the energy of

an electric current into heat energy. In this case the energy

of the current is said to be
consumed in overcoming the
resistance which the con-

ductor or the circuit offers
'^^^^^^ to its passage. Heat is de-
^^^' ^^^' veloped by a current in every

part of the circuit, because all substances offer some resist-
ance to a current ; in other words, there are no perfect
conductors. The small platinum wire offers much greater
resistance than an equal length of a larger copper wire ;
whence the greater quantity of heat generated in this part
of the circuit. All of the energy of any electric current that
is not consumed in doing other kinds of work is changed into
heat.

Fig. 38.3 represents a calorimeter of simple construction. An
inverted wide-mouthed bottle has its stopper pierced by two stout

484

ETHER DYNAMICS.

copper wires, which are united within tlie bottle by a coil of fine

platinum wire. Through a hole bored in the bottom of the bottle is

introduced a thermometer, T. The
bottle contains a known quantity of
water. Now if a current be passed
through the coil during a given time,
it is clear that the amount of heat
generated can easily be determined
by multiplying the known mass of
water by its temperature change, as
indicated by the thermometer. This
heat, however, is equivalent to a
definite quantity of electric energy
transformed in the wire.

By means of apparatus of tMs
kin^ Joule's law was established :
viz. The number of units of heat gen-
erated in a conductor varies as (1) its

resistance, (2) the square of the strength of the current, and (3) the

time the current flows.

450. (6) Physiological effects.

Experiment 9. — Place the copper electrodes of a single voltaic cell on
each side of the tip of the tongue. A slight stinging (not painful) sensa-
tion is felt, followed by a peculiar acrid taste.

Fig. 383.

Section V.

ELECTRICAL QUANTITIES AND UNITS.

451. Importance of electrical vieasicrements. — Less than
half a century ago the experimental sciences of electricity
and magnetism were in a great measure collections of iso-
lated qualitative results. Now, happily, all this has been
changed. The introduction of the absolute system of units
has been largely instrumental in changing experimental elec-
tricity and magnetism into sciences of which the most delicate
and exact measurement is the very essence.

STRENGTH OF CURRENT. 485

The wonderful developments which have been made in
recent years in electrical science, and which have led to the
employment of electric energy in connection with a great
diversity of industrial arts, are almost wholly due to a better
understanding of what electrical measurements can be made,
and how to make them. Indeed, little of a liractical nature
can be done without some acquaintance with the methods of
making these measurements.

452. Strength of current. — The expression strength of cur-
rent means the rate of flow of electricity. The "size" of a
stream of water, or the " rate of flow " might be indicated by
stating the number of gallons which flow past a given point
in a minute. We have not adopted in hydraulics any par-
ticular name for a gallon a. minute, but there is a necessity in
electricity for a term to denote the corresponding idea ; in
other words, there is a necessity for a unit for measuring
rate of flow or current strength.

The so-called C.G.S. electro-magnetic unit^ of current strengtli
is determined as follows : (First it is necessary to define a mag-
netic pole of unit strength. If a long thin magnet be broken in
the middle, the broken ends develop two opposite poles of equal
strength. Suppose these poles to be placed one centimeter apart
and the attraction between them to be measured by a very delicate
spring balance. If these poles are of such strength that at a dis-
tance of one centimeter they attract each other with a force of one
dyne, they are said to be of unit strength.)

Suppose that a thin wire is bent into a circle of 1 cm radius (Fig.
384), and a magnetic pole of unit strength is held at its center. As
we shall see further on, there is a force tending to move the mag-
netic pole along the lines of force of the magnetic field developed
by the current, i.e. in a direction at right angles to the plane of
the cncle. Let this force be measured in dynes, which can be

iThe units which we are here discussing are called electro-magnetic units to
distinguish them from units of a different nature called electrostatic units (see
Section VI), which are derived from the effects of electrostatic attraction and repul-
sion. The electrostatic units are chiefly used in connection with electrostatics, and
the electro-magnetic units in connection with electrodynamics. Of the electro-
magnetic units there are two systems, the C.G.S. and the practical.

486

ETHER DYNAMICS.

done by weighing in grams and multiplying the number of grams
by 980 (the value of g). Then since the length of the circular
current is 2 7rcm, the attraction of a unit length
2 of it is obtained by dividing the total force in

dynes by 2 7r. If now there be passed through
the wire a current of such a strength that this
force per unit of length is equal to one dyne, the
current is called a unit current.

A C.G.S. electro-magnetic unit of cur-
rent strength is defined to be the strength
of a current such that a centimeter of it acts
on a unit magnetic pole with a force of one
dyne, every point of the current being at a
distance of one centinfieter from, the pole.

We have found in dynamics that, in
practical work, the C.G.S. units are seldom
used on account of their inconvenient size.
The same applies to electricity. For prac-
tical use there was adopted by a Congress
of Electricians a class of units called the practical units, which
are certain multiples or submultiples ^ of the C.G-.S. units.

The practical unit of current strength, called the ampere, is
one-tenth of the C.G.S. unit. The quantity of electricity
conveyed per second by a current whose strength is one
ampere is called one coulomb. The coulomb is the practical
unit of quantity of electricity.

453. Electro-motive force. The volt. — Water flows from
one place to another in virtue of a difference of pressure be-
tween the two places, and the flow takes place from the place
of high pressure to the place of low pressure. For instance,
when water flows from a reservoir or cistern the pressure at
any point in the pipe is due to the " head " of water above it.

Fig. 384.

1 These practical units are derived from the C.G.S. electro-magnetic ones by-
adopting a new unit of length, the earth's quadrant (lO^ cm), and a new unit of
mass, 10—11 g, the second being retained as a unit of time.

ELECTRO-MOTIVE FORCE. 487

If it be set flowing by a force pump, we might say the flow of
water was due to a water-motive force which could be ex-
pressed as equal to a " head " of a certain number of feet of
water.

Similarly, electricity flows in a conductor only when there
is a difference of what may be termed electrical lyressure be-
tween its ends. If such be maintained between two points
connected by a conductor, it obviously represents a kind of
curvent-prodiicing force, one which can keep electricity in
motion against resistance. It is for this reason called electro-
motive force (E.M.F.). Electro-motive force is that which
maintains or tends to maintain a current of electricity through
a conductor. That which hinders the current is called re-
sistance.

Difference in electrical pressure we have hitherto assumed
to be due to difference of potential. It is this difference of
electrical pressure which sets up a current in the conductor.
Potential difference may be due to contact of dissimilar sub-
stances, as in the voltaic cell, or to the movement of a part
of the conductor in a magnetic field, as in the dynamo (§ 525).

Experience shows that if electricity be made to move in oppo-
sition to E.M.F., or, to speak figuratively, be carried up hill,
such a displacement of electricity against electrical stress requires
an expenditure of energy ; that is, it cannot be done without doing
work and drawing upon a supply of energy of some form. Dif-
ference of potential is therefore measured by the work done in con-
veying a unit of electricity in opposition to E.M.F. The C.G.S.
unit difference of potential is said to exist between two points where
one erg of work must be done in conveying a C.G.S. unit of elec-
tricity from one point to the other.

The volt is the name chosen for the practical unit of E.M.F.
and difference of potential, and this unit is equal to 10^
C.G.S. units. For purposes where great accuracy is not
required, it will answer to consider a volt as the E.M.F. of
a Daniell's cell ; i.e. it is aboict the difference of potential be-

488 ETHER DYNAMICS.

tween the zinc and copper of this cell, the E.M.F. of a
standard Daniell cell being approximately 1.07 volts.

454. Electrical luork and electrical activity. The joule OAid
watt. — The volt is of snch a magnitude that one conlomb of
electricity conveyed against an E.M.F. of one volt requires
an expenditure of one joule = one volt-coulomh of energy. The
volt-coulomb is analogous to the foot-pound and kilogram-
meter. Hence if a coulomb of electricity flow between two
points in a conductor whose difference of potential is one
volt, then one joule of work is done thereby.

If a conductor be traversed by a current of one ampere, i.e.
a coulomb per second, and we find two points whose difference
of electrical level is one volt, then the rate at which work is
being done in that portion is one tvatt = one joule per second.
The joule and watt are units of electrical work and electrical
activity, respectively.

455. Resista7ice. The ohm.

In every case in which a steady electric current flows in a con-
ductor, it is found that if tlie difference in potential between any
two points in the conductor be measured in volts, and this number
be divided by the strength of the current in amperes, a quotient
is obtained which has a constant value. That is, if the difference
of potential be doubled, the flow will be doubled, and so on. This
constant ratio of electric pressure to electric flow is called the
" electrical resistance of the conductor ; that is,

resistance = pressure -i- flow.

If pressure be measured in volts, and flow in amperes, the resist-
ance of a conductor in which a difference of pressure of one volt
produces a flow of one ampere is called one ohiii^^ and we have

1 Summary of Practical Electrical Units.
Names embalmed in scientific nomenclature.
English. French. German. Italian. American.

James Watt. Charles A. Coulomb. G. S. Ohm. A. Volta. J. Henry.

James P. Joule. Andre M. Ampere.

Names of units called after the above.
Poioer, the watt. Quantity, the coulomb. Resistance, the ohm. Self-induction,
Work, the joule. Cxirrent, the ampere. Pressure, the volt. the henry.

RESISTANCE. 489

pressure in volts -!- flow in amperes = resistance in ohms ;

or, divide the difference in potential between any two points by the
strength of the current, and the quotient is the resistance between
those two points in ohms. Hence resistance may be defined as the
ratio of the E.M.F. to the current strength.

The unit of resistance is called the ohm. Every substance
offers resistance to the passage of a current. Those sub-
stances which offer an immensely powerful barrier are called
insulators. Yet all substances conduct to some extent ; and
when an insulator is spoken of the term is only relative.

The ohm,^ as defined by the Paris Congress of Electricians
(1884), is the resistance ojfered by a column of pure mercury at
0° C, 1 sqiuvre millmieter in section and 106 centimeters longp-
It is about the resistance of 9.3 feet of No. 30 (American
gauge) copper wire (.01 in. diam.) at ordinary temperature.

The dimensional equation for quantity of electricity is

(Q)=[M^Li].

1 A megohm is one million ohms. A microhm is one millionth of an ohm.

2 In July, 1894, an act of U*. S. Congress Avas passed " To define and establish the
units of electrical measure." The following are quotations from this act. " The
unit of resistance, known as the international ohm, shall be represented by the
resistance offered to an unvarying electric current by a column of mercury at the
temperature of melting ice, fourteen and four thousand five hundred and twenty-one
ten thousandths grams in mass, of a constant cross-sectional area, and of the length
of one hundred and six and three-tenths centimeters."

" The unit of current shall be known as the international ampere and is the
practical equivalent of the unvarying current, which, when passed through a solu-
tion of nitrate of silver in water in accordance with standard specifications, deposits
silver at the rate of one thousand one hundred and eighteen millionths of a gram
per second."

The international volt "Is the electro-motive force that, steadily applied to a
conductor whose resistance is one international ohm, will produce a current of an
international ampere."

The international coulomb " Is the qiiantity of electricity transferred by a cur-
rent of one international ampere in one second."

The international " Unit of work shall be the Joule, which is practically equiv-
alent to the energy expended in one second by an international ampere in an inter-
national ohm."

The international unit of activity " Shall be the Watt, which is practically equiv-
alent to the work done at the rate of one Joule per second."

490 ETHER DYNAMICS.

The dimensional equation for current strength is

(V) = [M^L*T-2].
The dimensional equation for resistance is
(R) - [LT-i].

456. OJim^s Laiv. — The three factors, current (C), electro-
motive force (E), and resistance (E), are evidently inter-
dependent. Their relations to one another are stated in the
well-known Ohm's Law thus : The current is equal to the
electro-Tnotive force divided hy the resistance ; or

Hence the strength of a current is directly proportional to
the E.M.F. and inversely proportional to the resistance.
This famous law is at the base of a large portion of electrical
measurements, and its applications are developed in treatises
on the mathematics of the subject.

457. Resume. — The ampere is analogous to the "miner's
inch" used by miners and irrigators in the Western States.
The latter denotes the rate of flow of water which, under a
head of six inches, will pass through a hole one inch square
in a board two inches thick. Let this head of water represent
a volt, and the resistance of the hole one ohm; then the
miner's inch would represent a current of one ampere. The
expression "one miner's inch per second " is redundant; so
is the expression " one ampere per second."

A unit current is a current maintained by a unit E.M.F.
against a unit resistance.

A unit E.M.F. is the E.M.F. required to maintain a unit
current against a unit resistance. A conductor has a unit
resistance when a unit E.M.F. or a unit difference of poten-

ELECTKOSTATIC UNITS. 491

to pass

tial between its two
through it.

ends

causes

a unit curreni

The ampere
The volt
The ohm

= 10-1

= 108
= 109

C.G.S.

units of current.
" E.M.F.
" resistance.

From the numerical values given above it will be seen that if we
have a circuit in which the resistance is one ohm and the E.M.F.
one volt, then the strength of the current will be one ampere ; for

108

^^ = 10-1

109

Section VI.

ELECTROSTATIC UNITS.

Eeference has been made in the foregoing pages to various elec-
trical quantities, and we give below definitions of certain electro-
static units, rather as a convenience to the student for reference
than for practical use in connection with this work.

The electrostatic units embrace the units of quantity, potential,
and capacity. No names have yet been adopted for these units.

458. Unit of quantity.

One absolute unit of electricity is that charge on a very small
body which, if placed at a distance of one centimeter from an equal
charge, will exert through air a force of one dyne.

We can express the force between two quantities q and q' con-
densed in points d cm apart by —- (Compare with the law of
gravitation, §96.) The dimensional is g = [M*L^T-i].

459. Unit difference of potential.

Since potential represents work done on a unit of electricity, a
unit difference of potential may be defined as such a difference
of potential between two points as requires the expenditure of one
erg of work to transfer a + unit of electricity from one point to
the other, that point being at higher potential to which the + unit is
carried. The dimensional of difference of potential is d = M^L^T-i.

492 ETHER DYNAMICS.

460. Unit of capacity.

Since capacity is quantity per unit potential, a unit of capacity
is defined as such a capacity of a conductor as requires a charge of
one unit of electricity to raise it to unit of potential. Capacity,

c, is measured by ^, and its dimensional is c = L.

461. Electric force and intensity of electric field at a poi7it.

The electric force at any point in an electric field or the intensity
of an electric field at any point is the force with which a unit of
+ E would be acted on if placed at that point. Its dimensional, /,

is / = [M L T-2] -f [M* L"'" T-i] , or [M* L"* T-i] .

462. Specific inductive capacity.

The specific inductive capacity of a dielectric is the ratio of the
capacity of a condenser, the space between the plates of which is
filled with the dielectric, to the capacity of a precisely similar con-
denser with air as a dielectric. It is therefore simply a numerical
coefficient.

Section VII.

RULES RELATING TO AN ELECTRIC CURRENT.

463. Activity of a current. — The unit of electric activity
(or rate of doing work) is the ivatt. A watt is the activity of
a current of one ampere maintained by a difference of poten-
tial of one volt. 1 volt X 1 ampere =^ 1 volt-ampere or watt.
Hence

(1). A (watts) = C (amperes) X E (volts).

(2). The watt =c= (10- ^ X 10'^ =) 10^ ergs per second, or

— — horse-power. Hence = activity in horse-power.

For example, to find the rate at which energy is trans-
formed in an electric lamp, measure the whole current in am-
peres ; measure the difference of potential (with a voltmeter)
between the terminals of the lamp, in volts ; multiply together

ACTIVITY OF A CURRENT. 493

the quantities thus obtained and divide by 746 ; the result
will be the horse-power absorbed in the lamp. That is, a
current of C amperes falling E volts will perform, in passing

through the instrument, work at the rate of horse-power.

746

a. Substituting in equation (1) the value of C as given in

E E^

Ohm's formula, C= — , we have, A=— ; i.e. the activity is

equal to the square of the E.M.F. divided by the resistance.

&. E = CR (Ohm's formula). If this value of E be substi-
tuted, formula (1) becomes A := C^E ; i.e. the activity is di-
rectly x^'i'oportioiial both to the square of the current strength
when R is constant and to the resistance when C is constant.

(3). The amount of chemical decompositio7i produced by a
current in a given time varies as the strength of the current.
On this principle is constructed the voltameter, which measures
the strength of a current by the amount of chemical action it
effects in a given time.

(4). The mass in grams of an element deposited by elec-
trolysis is found by multiplying its electro-chemical equivalent
(i.e. the mass in grams of the element deposited by one am-
pere in one second) by the strength of the current in amperes,
and this product by the time in seconds during which the cur-
rent electrolyzes.

(5). The number of units of heat developed in a conductor is
pjroportional (1) to its resistance, (2) to the square of the strength
of the current, and (3) to the time the current is flowing.

A current of one ampere flowing through a resistance of
one ohm develops therein 0.00024 calorie of heat per second.
Hence H (calories) = C^ (amperes) X R (ohms) X t (seconds)
X 0.00024.

Whenever the current heats a wire, produces decomposition,
or performs work of any kind, each of these acts is accom-
plished at the expense of the potential energy in the battery.

494 ETHER DYNAMICS.

If the current operate an electric motor which pumps water,
or lifts a hammer, the battery loses energy proportional to the
work required for each of these mechanical acts.

Section VIII.

INSTRUMENTS FOR ELECTRICAL MEASUREMENTS.

Our attention is next directed to the consideration of in-
struments for measuring the quantities which, we have seen,
are required to be known. First we consider the instrument
for measuring quoMtity of electricity, properly called a coulomb-
meter.

464. Coulomh-m&ter. Voltameter. — The simplest quantity-
meter is based on the electrolytic effect of the current, and is
called a voltameter. If, for instance, using the Hoffman vol-
tameter (Fig. 368), we measure the hydrogen generated during
a given time, the mass or volume of this hydrogen under con-
stant pressure and temperature is exactly proportional to the
number of coulombs of electricity which have passed through
the liquid. The mass in grams of any constituent of an elec-
trolyte liberated by the passage of one coulomb of electricity,
is called its electro-chemieal equivalent. For commercial pur-
poses, instead of the coulomb as the unit of quantity, a larger
unit, the amxjere-hour (^3600 coulombs), is frequently used.

The following are the electro-chemical equivalents ^ per
coulomb and per ampere-hour for a few metals : —

Electro-chemical Equivalent per

equivalent per coulomb. ampere-hour.

Hydrogen ; . . .000010354 grams .03727 grams.

Silver 00111800 " 4.0248

Copper 0003284 " 1.1822

Zinc 00033696 " 1.223056 . "

Lead 00107160 " 3.85776

1 Tlie student in chemistry will understand that the electro-chemical equivalents
of different metallic elements are proportional to their combining equivalents.

GALVANOMETER. 495

Edison in his system of electric lighting employs a zinc
voltameter for measuring the quantity of electricity furnished
to each customer. It consists of two zinc plates immersed in
a solution of zinc sulphate, and is so arranged by means of a
divided circuit (§ 477) that only a portion (say a thousandth)
of the current passes through the liquid. The increase of
mass of the electro-negative plate in grams divided by 1.223
gives the quantity in ampere-hours which has passed through
the voltameter.

In like manner the electrolytical action in the voltaic cell
itself is proportional to the strength of the current while it
passes. One coulomb of electricity in passing through a
Daniell's cell dissolves .00033696 gram of zinc and deposits
.0003284 gram of copper.

A coulomb-meter will serve as an ampere-meter (abbreviated,
ammetery when the current is very nearly constant, but not
otherwise. Eor the quantity of electricity measured in
coulombs which has passed through the circuit in a given
time, divided by the number of seconds, obviously gives the
coulombs-per-second, or the mean ampere strength of the
current.

465. Galvanometer. — This is an instrument for measuring
current-strength by means of the deflection of a magnetic
needle when placed in the field of the current. It is so con-
structed that either the deflection angle itself, or some
function of it, is proportional to the current-strength.

466. Thompson's mirror galvanoTneter. — A simplified form
is shown in Fig. 385, and the complete instrument is shown
in Mg. 386. Insulated wire is wound on a bobbin, A. With-
in this bobbin is hung, by a silk fiber, a little circular concave
mirror, to the back of which are attached little magnets of
watch-spring steel. To adjust it for use, the suspended
magnets must be set parallel to the coil of wire. For this
purpose it is necessary to have a small controlling magnet,

496

ETHER DYNAMICS.

n s, to cause the needles to take the required position. The
galvanometer is also rendered more or less sensitive by moving
the controlling magnet farther from or nearer to the needles.

Fig. 385.

If a beam of light from a lamp in a dark room be thrown
upon the mirror and reflected thence to a screen, S, a spot of

Fig. 386.

light thereon will show the slightest change in the position
of the mirror-needle in the coil. Hence a very feeble current

TANGENT GALVANOMETER.

497

declares its presence by causing the spot of iiglit to move to
one side or the other of a central or zero point where the spot
falls when there is no current. With this galvanometer no
appreciable error is committed in considering the current
strengths as proportional to the scale-readings. This instru-
ment is of great value to the electrician in dealing with very
weak currents.

467. Tangent galvanometer. ■ — ■ A tangent galvanometer is
one so constructed that the current passing through it is pro-
portional to the tangent of the angle of deflection produced.
To this end it is necessary that the needle be very short (not
more than gL) in comparison with the diameter of the coil.

In its simplest form it consists of a large vertical coil
(better two coils, one on each side of the needle, Fig. 387,
so placed that the needle is
at the center of the common
axis), in the center of which
is either a small compass
needle or a needle suspended
by a silk fiber.

A needle thus placed in the
field of a current is acted on
by a mechanical couple tend-
ing to place it at right angles
to the plane of the coil, and it
is deflected until this couple is
balanced by the return couple
due to the earth's magnetism.
The value of the earth's mag- ^'''- ^^^•

netic intensity in a horizontal plane is denoted by H.

The formula for the tangent galvanometer is C =

2 7rn

tan

Ktan 6, in which C is the current strength in C.G.S. units ; H, as
above, measured in dynes ; 6, the angle of deflection ; and r, the

498 ETHER DYNAMICS.

mean radius of the coil of n turns. K, which is called the reduction
factor of the galvanometer, is usually written in the form

2wn G
r
It is made up of two patts, viz. H, which is dependent on locality,
and G, which depends on the construction of the instrument, and is
therefore called the galvanometer constant. Hence, if the value of
H and G be once found, the strength of any current is calculated
by multiplying the tangent of the deflection angle by the ratio K.
As an ampere = 10— i C.G.S. unit, the current strength in amperes
is found by multiplying this value by 10.

When the scale is divided into degrees, the corresponding tan-
gents are found by consulting a table of tangents (p. 626). In some
instruments, however, the scale is graduated directly in tangents.

The process of finding the value of the reduction factor of any
instrument is called standardizing. There are many ways of doing
this, which may be found in any good laboratory manual. One of
these consists in introducing a copper voltameter into circuit with
the galvanometer. After passing a current for a certain time, and
observing the deflection of the galvanometer during that time, ascer-
tain the gain of mass in grams of the negative plate, and divide this
gain by the time in seconds. This gives the deposit of copper per
second. Dividing this result by .0003284 (§ 464) gives the ampere
current which passed through the galvanometer. Finally, divide
the last result by 10 tan 6 and the result is the value of K. That
is, the reduction factor of a galvanometer = strength of current in
amperes -r 10 tan 6, when 6 is the average deflection, i

It has been found that errors of observation affect the value of C
least when the mean deflection is 45° ; hence, it is customary to
arrange the experiments so that about this deflection angle may
be produced.

If the strengths of two currents are to be compared, it is
only necessary to obtain deflections with each current sepa-
rately, and compare the tangents of the angles.

468. Ammeter. — Now if the value of each division of the

1 It is to be borne in mind tliat the formula C = K tan measures the current in
C.G.S. electro-magnetic units and not in amperes ; and that 1 C.G.S. unit is equal to
10 amperes.

AMMETER.

499

scale be found by multiplying the number indicating the
tangent of the angle of deflection by 10 K, and these results
be placed iipon the scale in place of degree numbers, we shall
have a direct-reading ampere-meter (ammeter). There is a
great variety of ammeters in use, for a description of which
the student is referred either to technical works on the sub-
ject or to the inventors themselves.

We shall consider only one other form, that called the
(Kohlrausch) solenoid ammeter, selecting this because of its
simplicity. In Eig. 388, a is a helix of thick wire, ^'is a soft
iron tube which serves as a core, suspended by a light spring

Fig. 388.

c. The core carries a marker ^y / is merely a wooden guide-
rod for the tubular core. The action of this ammeter depends
upon the principle that when an insulated wire is wound into
a helix (called also a solenoid), and a current is passed
through the wire, an iron rod or tube placed at the opening
will be drawn into the helix with a force increasing with the

500 ETHER DYNAMICS.

strength of the current. This force acting against the elastic
force of the spring may be measured in the same manner as
weight by a spring balance. Now as the current strength
bears a definite relation to the force, the instrument can
easily be calibrated in amperes.

Section IX.

RESISTANCE OF CONDUCTORS.

469. External and internal resistance. — For convenience
the resistance of an electric circuit is divided into two parts,
the external and the internal. External resistance includes
all the resistance of a circuit except that of the generator,
while the latter is termed internal resistance.

When the external resistance in a circuit is considered
separately from the internal, Ohm's formula must be con-
verted thus (calling the former R, and the latter r): —

If a cell have E = 1 volt, and r = 1 ohm, and the connecting
wire be short and stout, so that E may be disregarded, then
the cell yields a current of one ampere. If by any means
the internal resistance of this cell can be decreased one-half,
it will then be capable of yielding a two-ampere current
under the same conditions.

470. External resistance.

Experiment 1. — Introduce into a circuit a galvanometer, i and note
the number of degrees the needle is deflected. Then introduce into the

1 The galvanometer represented in the cut is a form of galvanometer chiefly used
by the author in elementary laboratory work. The results obtained by it are ap-
proximately those which would be obtained by a standard tangent galvanometer.
The manifold uses to which galvanometers are put in a physical laboratory properly
require a variety of instruments, and this would make a complete equipment qxiite
expensive.

EXTERNAL RESISTANCE.

501

same circuit the wire on tlie spool numbered 4 on tlie platform, 2 S
(Fig. 389). (The wire on any one of the five spools on this platform can
at any time be introduced into a circuit, by connecting the battery wires
with the binding screws on each side of the spool to be introduced.)

Pig. 389.

The deflection is now less than before. The copper wire on this spool
is 16 yards in length ; its size is No. 30 of the Brown and Sharpe wire
gauge. When this spool is in circuit, the circuit is 16 yards longer than
when the spool is out. The effect of lengthening the circuit is to weaken
the current, as shown by the diminished deflection.

Experiment 2. — Next, substitute Spool 2 for Spool 4. This contains
32 yards of the same kind of wire as that on Spool 4, The deflection is
still smaller.

The weakening of the current by introducing these wires is caused by
the resistance which the wires offer to the current, much as the friction
between water and the interior of a pipe impedes, to some extent, the
flow of water through it. The longer the pipe the greater is the re-
sistance to the flow.

If the wire on the spools had been the only resistance in the circuit,
then, when Spool 2 was in the circuit, the resistance would have been
double what it was when Spool 4 was in the circuit, and the current,
with double the resistance, would have been half as strong.

2 The platform of spools containing wire of different (known) sizes, lengths, and
material, so arranged that any one, two, or more can he introduced into the circuit
for the purpose of measurement of resistance, is an instrument of great convenience
in a school laboratory.

502 ETHER DYNAMICS.

(1) Other things being equal, the resistance of a conductor
varies as its length.

Experiment 3. — Next substitute Spool 1 for Spool 2. This spool
contains 32 yards of No. 23 copper wire, — a thicker wire than that on
Spool 2, but the length of the wire is the same. The deflection is now
greater than it was when Spool 2 was in circuit. This indicates that the
larger wire offers less resistance.

Careful experiments show that (2) the resistance of all
conductors varies inversely as the areas of their cross sections.
If the conductors he cylindrical it varies inversely as the square
of their diameters.

Experiment 4. — Substitute Spool 5 for Spool 1 , and compare the
deflection with that obtained when Spool 4 was in the circuit. The de-
flection is smaller than when Spool 4 was in circuit. The wire on these
two spools is of the same length and size, but the wire of Spool 5 is Ger-
man-silver. It thus appears that German-silver offers more resistance
than copper. ,

(3) In obtaining the resistance of a conductor, the specific
resistance of the substance must enter into the calcidation.
(See table of specific resistances in the Appendix.)

The resistance of metal conductors increases slowly with
the temperature of the conductor? The resistance of German-
silver is affected less by changes of temperature than that of
most metals ; hence its general use in standards of resistance.

471. Internal resistance.

Experiment 5. — Connect with the galvanometer the copper and zinc
strips used in Experiment 1, Section 1, and introduce the strips into a
tumbler nearly full of acidulated water. Note the deflection. Then raise
the strips, keeping them the same distance apart, so that less and less of
the strips will be submerged. As the strips are raised, the deflection be-
comes smaller. This is caused by the increase of resistance in the liquid
part of the circuit, as the cross section of the liquid lying between the
two strips becomes smaller.

DESCRIPTION OF THE RESISTANCE BOX.

503

(4) The internal resistance of a circuit, other things being
equal, varies inversely as the area of the cross section of the
liquid between the two elements.

In a large cell the area of the cross section of the liquid
between the elements is larger than in a small cell, con-
sequently the internal resistance is less. This is the only
way in which the size of the cell affects the current.

Obviously the resistance of the battery would be increased
by any increase of the distance between the elements, since
this increases the length of the liquid conductor, but as this
distance is usually made as small as convenient, and is kept
invariable, it demands little of our attention.

Section X.

MEASUREMENT OF KESISTANCE.

472. Description of the resistance box.

Fig. 390 represents a wooden box containing what is equivalent to a
series of coils of German-silver wire, whose resistance ranges from 0. 1 ohm
to 100 ohms.i Each of
these coils is connected ^ A .

with a brass stud on the
top of the box.

Three switches, A,
B, and C, so connect
the coils with the bind-
ing screws a and 6 that
a current can be sent
through any three coils
at the same time by
moving the switches on
to the proper studs.

The resistance in ohms of each coil is marked on the box near its stud.
When the three switches rest upon studs marked 0, the current meets
with no appreciable resistance in passing through the box, but any

1 Each additional switch with its corresponding coils increases the range about
tenfold, so that the range of the instriiment may be very much increased.

Fig. 390.

504

ETHER DYNAMICS.

Fig. 391.

desired resistance within the range of the instrument can be introduced
by moving the switches on to the studs, the sum of whose resistances is
the resistance required. This instrument we shall call a resistance box.

473. Wheatstone bridge.

Fig. 391 represents a perspective view of the bridge (as modified
by the author), and Eig. 392 represents a diagram of the essential
electrical connections. ^ The battery wires are connected with the
bridge at the binding screws B B'. A galvanometer, G^ is con-
nected at GG', a resist-
ance box, r, at K E, and
the conductor x, whose
resistance is sought, at
XX.

When the circuit is
closed by means of the
key T, the current, we
will suppose, enters at B ;
on reaching the point A it divides, one part flowing via the branch
A G B', and the other via the branch A D B'. If points D and G
in the two branches be at
different potentials and a
connection be made be-
tween them through the
galvanometer, (9, by clos-
ing the key S, there will
be a current through this
wire and through the gal- /qq^^-
vanometer, and a deflec ^—^
tion of the needle will be /Q CIh-
produced. But if the
points D and G be at
the same potential, there
will be no cross current
through the bridge wire
and no deflection. Now
it can be demonstrated
that points D and G will be at the same potential when R (the re-

1 The student will find descriptions of the more elaborate bridges and resistance
coils in such works as Gordon's Electricity and Magnetism, Vol. I, Sylvanus Thomp-
son's Lessons in Electricity and Magnetism, and in various laboratory manuals.

/es

WHEATSTONE BRIDGE. 505

sistance) of A D : R of D B' : : R of A G : R (the unknown resistance)
of G B'. Between A and D and A and G there are three coils of
wire having resistances respectively of 1, 10, and 100 ohms. One or
more of these coils are introduced into the circuit by removing the
corresponding plugs a, &, c, cZ, e, and /. As the other connections
between A and D, and A and G, have no appreciable resistance,
being for the most part short brass bars, the only practical resist-
ance between these points is that introduced at will through the
coils. Similarly between points D and B^, the only practical resist-
ance is that introduced at will through the resistance box, and
between the points G and B' the resistance is the resistance (x)
sought.

It is apparent, then, that in using the bridge after the connec-
tions are properly made through the several instruments and certain
known resistances are introduced between A and D, and A and G,
we have simply to regulate the resistance through the resistance
box so that there will be no deflection in the galvanometer ; then
we are sure that the above proportion is true. The first three
terms of the proportion being known, the fourth term, which is the
resistance sought, is computable. i

If the same resistance be introduced between points A and
G as between A and D, it is evident that the resistance in
the -resistance box r must be made equal to the unknown
resistance x in order that there may be no deflection in the
galvanometer. Consequently when this result is obtained
the resistance of x may be read from the resistance box.

Experiment. — Measure the resistance of each of the several spools of
wire used above, — electro-magnets, electric lamps, etc., — using the
bridge. Place the switches of the resistance box on the zero studs.
Make connections as in the description above. Then close the circuit at

1 The accuracy of the results obtained largely depends upon so choosing resist-
ances of the bridge as to make the arrangement have maximum sensibility, and upon
the sensitiveness of the galvanometer. In using the bridge the following directions
should be observed : (1) Always close the circuit at T before closing the bridge at S,
and in breaking the circuit reverse this order. (2) Introduce between A and D, and
A and G, resistance as nearly equal to the resistance sought (a;) as practicable. If
you have no conception what the unknown resistance is, it is best to begin by using
high resistances. (3) Use a sensitive galvanometer, e.g. a mirror galvanometer, or
the galvanometer shown in Fig. 389, substituting the astatic needle for the tangent
needle.

506

ETHER DYNAMICS.

T, and afterwards the bridge at S. There will probably be a deflection
in the galvanometer. Regulate the resistance through the resistance
box, throwing in or taking out resistance according as one or the other
tends to reduce the deflection (the process is much like that of weighing),
until there is no deflection. Then compute the resistance sought accord-
ing to the above proportion.

474. Measurement of galvanometer resistance. Lord Kelvirv's
method. — The bridge may be used for measuring the resist-
ance of the galvanometer actually in
use. The bridge is arranged as in
Fig. 393. The resistance in the re-
sistance box K is then varied until
the deflection of G does not change
when the key S is closed ; then

a
r = K-y

in which r is the resistance of the
galvanometer, K is the resistance in
the resistance box, and a and h are the
resistances in the arms A G' and A D
respectively. If a = h, then r = R.

475. Battery resistance. Mance's Tnethod.

No definite meaning can be attached to the expression battery
resistance, since this resistance is complicated by variation in the
polarization, and this in turn is dependent upon the strength of the
current, external resistance, etc. Hence the numerous methods
depending on Ohm's law that have been devised for deducing battery
resistance are of little value in many cases.

The method known as Mance's method is one of the best for
measuring battery resistance, since it requires the battery to be
constant only during the short time the key is closed. A Wheat-
stone bridge is arranged as shovm in Fig. 394. The resistance in
the resistance box E is adjusted until, on pressing down the key
A, the deflection of the galvanometer does not change ; then

r = R

ELECTRO-MOTIVE FORCE OF DIFFERENT CELLS. 507

in which r is the battery resistance, R is the resistance in the re-
sistance box ; a and h are the resistances respectively in the arms
A G' and AD. If a = 6, the for-
mula is simplified, and becomes
r = R.

The information generally re-
quired in practice, however, is
what available difference of po-
tentials can be obtained with a
certain working resistance in the
external circuit. This can be
obtained by connecting the ter-
minals of the battery by the
body offering this resistance, and
measuring the difference of po-
tentials between these points by
means of a potential galvanom-
eter. If we call this difference
of potentials V, and the E.M.F.
of the battery when on open circuit E, and R
then we may write

E ^ I ^ p

R + r R '

Fig. 394.

external resistance,

where r is such a quantity as satisfies this equation. In other
words, tills quantity may be taken as the resistance of the battery
for the current C.

Section XI.

E.M.F. OF DIFFERENT CELLS. DIVIDED CIRCUITS.
OF COMBINING VOLTAIC CELLS.

METHODS

476. Electro-motive force of different cells. — If a galva-
nometer be introduced into a circuit with different battery cells,
e.g. Bunsen, Daniell, G-renet, etc., very different deflections
will be obtained, showing that the different cells yield cur-
rents of different strengths. This may be in some measure
due to a difference in their internal resistance, but it is chiefly
due to the difference in their electro-motive forces. We

508 ETHER DYNAMICS.

learned (§ 431) that difference of electro-motive force is
due to the difference of the chemical action on the two plates
used, and this depends npon the nature of the substances
used. It is wholly independent of the size of the plates ;
hence the electro-motive force of a large battery cell is no
greater than that of a small one of the same kind. Conse-
quently any difference in strength of current yielded by
battery cells of the same kind, but of different sizes, is due
wholly to a difference in their internal resistances.

The electro-motive forces of the Bunsen, Daniell, and
Grenet cells are respectively about 1.8, 1, and 2 volts.

477. Divided circuits ; shunts.

Experiment 1. — Make a divided circuit as in Fig. 395 (using double
connectors a and 6). Insert a galvanometer, G, in one branch and a
resistance box, K, in the other. When the current
reaches a, it divides, a portion traversing one branch
through the galvanometer, and the remainder passing
through the other branch and the resistance box. The
branch a R 6 is called a sliunt or derived circuit. In-
crease gradualljT^ the resistance in the resistance box.
The result is that it throws more of the current
through the galvanometer, as shown by the increase of
Fm. 395. deflection.

In a divided circuit the current divides between the paths
inversely as their resistances. For example, if the resistance
of the resistance box above be 4 ohms, and the resistance in
the galvanometer be 1 ohm, then four-fifths of the current will
traverse the latter and one-fifth the former.

Suppose that the resistance box and galvanometer be
removed from the shunts, and that the shunts be of the
same length, size, and kind of wire, and consequently have
equal resistances. Using the two wires instead of one to
connect a and h is equivalent to doubling the size of this
portion of the conductor ; consequently the resistance of this
portion is reduced one-half.

KIllCHHOFF S LAW.

509

Generally, the joint resistance of two branches of a circuit is
the product of their respective resistances divided hy their sum.
478. Kirchhoff's law.

The following discussion will make the above law evident. First
it should be understood that when
a conductor conveys a constant ' j

current the strength of current
across all cross - sections of the
conductor, as A, C, E F, D, and
B (Fig. 396), is the same. Hence
the current arriving at C or D of
the main circuit is equal to the
sum of the currents which flow
by the branches ri, r^ and r^-
This is known as Kirchhoff's First
Law.

By Ohm's law if two points, a
and &, between which a differ-
ence of potential Y is maintained,
be connected by two wires havin^

V V

current in that of resistance r i will be — ' and in the other — '

If C be the whole current flowing in the circuit, we have by
Kirchhoff's law

V V V

c=- + - = ^'

ri To R

where R is the resistance of a wire which might be substituted for
the divided conductor between a and 5 without affecting the cur-
rent. Hence

\ = — ' and R = — V^ "

Tx r-2 R ■ ri + Ti

resistances rx and r2, the

If there be three separate wires of resistance, as in Fig. 396, we
shall have in a similar manner

R Tx Ti Ts

The reciprocal of the resistance R of a wire, i.e. — ' is called its
conductivity, sometimes expressed as mhos.'^ We may say, there-

1 A word formed by writing the word ohin in reverse order.

510 ETHER DYNAMICS.

fore, in general, when two points in a circuit are connected by a
multiple arc (a term in common use to denote a divided circuit
between any two points) consisting of n brandies, the conductivity
of the multiple arc is equal to the sum of the conductivities of the n
branches : in other words, the reciprocal of its resistance is equal to
the sum of the reciprocals of the resistances of its branches.

479. Shunted galvanometer. — When a current is so strong
as to produce too violent an impulse upon tlie needle of a
galvanometer, its terminals may be shunted through a re-
sistance box, so that any known fraction of the current may
be deflected through the shunt.. For example, if the shunt
have a resistance i as great as that of the galvanometer, then
the current through the latter will be i that through the
shunt, or yL of the total current.

480. Methods of comhining cells.

Experiment 2. — Take two Bunsen cells, and connect the two zinc
plates by a wire. Then connect each of the carbon plates with a gal-
vanometer. The E.M.F. of each cell would tend to send a current
opposite to that of the other cell. But you find that there is either no
deflection in the galvanometer, or at most a very small one, and this
shows either that there is no current or that the current is very weak.
The reason is evident. You have connected two carbons, which have
the same potential, through the galvanometer ; consequently there should
be no current between them. The cells are said to be connected in
opposition.

A very simple way of showing that a large cell has no
greater electro-motive force than a small one is to connect
two such cells in opposition through a galvanometer, or,
what answers the same purpose, raise the zinc of one of two
cells of the same size, connected in opposition, nearly out of
the liquid. The absence of a current shows that the two
carbons have the same potential, and consequently their
electro-motive force is the same.

A number of cells connected in such a manner that the
currents generated by all have the same direction constitutes
a voltaic battery.

BATTERIES OF LOW INTERNAL RESISTANCE. 511

The object of combining cells is to get a stronger current
than one cell will afford. We learn from Ohm's law that
there are two, and only two, ways of increasing the strength
of a current. It must be done either by increasing
the E.M.F. or by decreasing the resistance. So
we combine cells into batteries, either to secure
greater E.M.F. or to diminish the internal resist-
ance. Unfortunately, both purposes cannot be
accomplished by the same method.

481. Battei'ies of loiv ■ internal resistance. ~Yig.
397 represents three cells having all the carbon (+)
plates electrically connected with one another, and
all the zinc ( — ) plates connected with one another,
and the triplet carbons are connected with the
triplet zincs by the leading-out wires through a
galvanometer G.

It is easy to see that through the battery the
circuit is divided into three parts, and consequent-
ly the conductivity in this part of the circuit, ac-
cording to the principle stated in § 478, must be
increased threefold ; in other words, the internal
resistance of the three cells is one-third of that of a single
cell. This is called connecting cells " in multiple arc," and
the battery is called a " battery of low internal resistance."
The resistance of the battery is decreased as many times as
there are cells connected in multiple arc, but the E.M.F. is
that, of one cell only.

The formula for the current strength in this case is written

Fig. 397.

thus

C =

E-f-

in which n represents the number of cells. It is evident

from this formula that when E is so great that - is a small

512

ETHER DYNAMICS.

part of the whole resistance of the circuit, little is added to
the value of C by increasing the number of cells in multiple
arc.

482. Batteries of high internal resista^ice and great E.M.F.
— Fig. 398 represents four cells having the carbon or + plate
of one connected with the zinc or — plate of the next, and

the -|- plate at one end of the

series connected by leading-out

wires through a galvanometer

with the —plate at the other

m I ■ I 1 I ■ ^^^^ ^^ *^^ series. It is evi-

1+ - 1+ " I"'' ~ I"*" ^^^^* ^^^^ *^^ current in this

■ ■ ' ■ series traverses the liquid four

times, which is equivalent to
lengthening the liquid conductor four times, and of course
increasing the internal resistance fourfold. But, while the
internal resistance is increased, the E.M.F. of the battery is
increased as unany times as there are cells in series. The gain
by increasing the E.M.F. more than offsets, in many cases
(always when the internal resistance is a small part of the
whole resistance of the circuit), the loss occasioned by in-
creased resistance.

The formula for current strength in this case becomes

U-\-7ir
It is evident that C is increased most by adding cells in
series when n r is smallest in comparison with E.

483. Best arrangement of cells.

Experiment 3. — Introduce into circuit witli a single cell a resistance
box and a galvanometer. Throw a resistance of (say) 50 ohms into the
circuit by means of the resistance box. Note the deflection. Then add
another cell, in series, to the cell already in use. The deflection is con-
siderably increased. Other cells -may be added with similar results.

Experiment 4- — Connect the two cells in multiple arc, keeping the

BEST AKRANGEMENT OF CELLS. 513

same resistance in the resistance box. The deflection is only a very little
greater than that caused by a single cell.

Experiment 5. — Connect a single cell with a galvanometer of low
resistance, so that the whole external resistance may be less than the
resistance of the single cell. Note the deflection. Then introduce an-
other cell in multiple arc. The deflection is considerably increased.

Experiment 6. — Connect the same cells in series. The deflection
differs but little from that produced by a single cell.

Hence, (1) wheii the external resistance is large, connect cells
in series ; (2) luhen the external is less than the internal re-
sistance, connect cells in multiple arc.

The two systems may be combined in one battery. Thus
one pair in series may be placed in multiple arc with another
pair in series. This combination would give double the
E.M.F. of a single cell, but the resistance of only one cell.

With a given number of cells and a given external resistance
the maxiniuin current is generated when the external and in-
ternal resistances are er[ual. It is seldom possible in practice
so to join the cells as to fulfill this condition ; but if the
strongest possible current be required it should be fulfilled
as nearly as possible. The fallacy, however, of introducing
resistance into any part of the circuit for the purpose of
making these resistances equal must be carefully avoided,
for resistance wherever introduced can tend only to w^eaken
the current. Nor must it be supposed that of two batteries
of equal E.M.F. , but one having a high, the other a low re-
sistance, the former is better adapted for working through a
high resistance. It should be borne in mind that the role of
a battery in general is to maintain a difference of potential
between its poles, and the element of resistance that it intro-
duces into the circuit is a necessary evil, not to be voluntarily
increased.

Electro-magnets and galvanometers must he adcipted to the
circuits in which they are to he placed. In connection with
the above discussion, it seems proper to introduce, somewhat

514 ' ETHER DYNAMICS.

parenthetically, a few facts pertaining to tlie use of electro-
magnets and galvanometers.

Li order to produce the greatest effect, the resistance of the
helix of an electro-magnet should be equal to that of the portion
of the circuit not included in the helix, i.e. to the rest of the
circuit. When several electro-magnets are used in the same
circuit, the sum of the resistances of all the helices should be
equal to the resistance of the rest of the circuit.

The same rule applies to galvanometers. High resistance
galvanometers are most suitable for high resistance circuits,
and loAv resistance galvanometers are most suitable for low
resistance circuits. In other words, both galvanometers and
electro-magnets should be adapted to the resistance of the
circuit in which they are to be used.

Exerciser.

1. What E.M.F. is required to maintain a current of one ampere
through a resistance of one ohm ?

2. Through what resistance will an E.M.F. of ten volts maintain a
current of 5 amperes ?

3. What current ought an E.M.F. of 20 volts to maintain through a
resistance of 5 ohms ?

4. A voltmeter applied each side of an electric lamp shows a differ-
ence of potential of 40 volts ; what current flows through the lamp, if it
have a resistance of 10 ohms ?

5. The resistance between two points in a circuit is 10 ohms. An
ammeter shows that there is a current strength in the circuit of 0.5
ampere ; what is the difference in potential between the points ?

6. What current will a Bunsen cell furnish when y = 0.9 ohm (about
the resistance of a quart cell), E = 1.8 volts, and R = 0.01 ohm (about
the resistance of 3 ft. of No. 16 wire) ?

[In the following exercises, whenever a Bunsen cell is mentioned it
may be understood to be a quart cell, having a resistance of about 0.9
ohm. Its E.M.F. is about 1.8 volts.]

7. a. AVhen is a large cell considerably better than a small one ?
h. When does the size of the cell make little difference in the current ?

8. If you have a dozen quart cells, how can you make them equiva-
lent to one 3-gallon cell ?

EXERCISES. 515

9. If a battery of 10 cells have an E.M.F. ten times greater than
that of a single cell, why will not the battery yield a current ten times
as strong ?

10. a. The internal resistance of ten cells, connected in multiple arc,
is what part of that of a single cell ? b. If the cells were connected in
series, how would the resistance o| the battery compare with that of one
of its cells? c. How would the E.M.E. of the latter battery compare
with that of a single cell ?

11. What current will a single Bunsen cell furnish through an external
resistance of 10 ohms ? #>

12. What current will 8 Bunsen cells, in series, furnish through the
same resistance ?

SoLOTiox : ^ = io!^^o^/^g = 0.83 + ampere.

13. What current will 8 Bunsen cells, in multiple arc, furnish through
the same external resistance ?

Solution: ^ = ^^^-iL_ = q. 17 + ampere.

14. What current will a Bunsen cell furnish through an external re-
sistance of 0.4 ohm ?

15. What current will a battery of two Bunsen cells, in series, furnish
through the same resistance as the last ?

16. What current will two cells, in multiple arc, furnish through the
same resistance ?

17. A coil of wire having a resistance of 10 ohms carries a current of
1.5 amperes. Eequired the difference of potential at its ends.

18. What would be the resistance at 0° C. of a column of mercury
154 cm long and f of a square millimeter in cross-section ?

19. a. What is the resistance of J mile of No. 16 (diam. .05 in.) copper
wire? b. What E.M.F. will be required to maintain a current of .5
ampere in this circuit ?

20. a. The resistance between two points, A and B, of a conductor is
2.5 ohms ; the resistance of a shunt between the same points is 1.5 ohms ;
what is the joint resistance between these points ? 6. If a current of 10
amperes be maintained between these points, what will be the strength
of current in each branch? c. How would the strength of current
between these points be affected if the shunt be removed and the same
fall of potential be preserved ? Why ?

21. a. Points A and B in a circuit are connected in multiple arc by
three branches whose respective resistances are 2, 3, and 4 ohms. State

516 ETHER DYNAMICS.

in order their relative conductivities, b. State the joint conductivity of
the multiple arc. c. State the joint resistance of the multiple arc.

22. Four conductors in multiple arc have resistances of 100, 50, 27,
and 19 ohms. What is their combined resistance ? Ans. 8.3 ohms.

23. Assume a current of 30 amperes and an E.M.F. of 50 volts ; what
is the resistance and conductivity ?

24. A wire is 40 mils (a mil is .001 in.) in diameter, 3 miles long, and
offers 40 ohms resistance. A second wire of the same material is 50 mils
in diameter and 9 miles long. What is the resistance of the latter ?
Ans. 76.8 ohms.

25. If the terminals of a galvanometer be shunted with a resistance ^^
that of the galvanometer, what part of the total current will the galva-
nometer measure ?

26. An electric lamp has a resistance of 50 ohms ; it is connected to a
street main by leads of 2^ ohms resistance. The heat developed in the
leads is wasted. What portion of the entire heat developed in the derived
circuit is wasted ?

27. a. The internal resistance of a voltaic circuit is 2 ohms, and the
external resistance is 16 ohms ; what portion of the entire heat developed
in the circuit is generated in the battery ? b. If the external resistance
of this circuit be reduced to .5 ohm, the heat generated in the battery
will be what part of the total heat developed in the circuit ?

28. What is the strength of a current which deposits .02 gram of
copper per minute ?

29. Suppose that there are a number of cells joined in series but the
circuit is completed by short, thick, copper leading wires of practically no
resistance, would any advanta,ge be gained by adding thereto more cells
in series ? Exx)lain.

30. A battery of 20 cells is divided into four groups. Each group
consists of five cells connected in series, and the four groups are con-
nected in multiple arc. Compare the E.M.F. and the resistance of this
battery with that of a single cell.

31. What E.M.F. of a dynamo generator will be necessary to maintain
a 12-ampere current for 100 arc lamps in series, each of which has a
resistance of 5 ohms, the resistance of the line wire being 20 ohms, and
the dynamo resistance being 25 ohms ?

CURRENT, RESISTANCE, POTENTIAL DIFFERENCE. 517

Section XII.

VERIFICATION OF OHM S LAW.

484. Relation of current, resistajice, and pote7itial difference.
This relation is best understood by the aid of the hydraulic

6^^-:^%

a
a'

lb E

d I

e
e |.

analogue. B (Fig. 399) represents a tank in which water is main-
tained at a fixed level by means of a pump (battery), while the
tank discharges through a pipe (conductor) N 0. At equal inter-
vals glass tubes, serving as manometers (potential meters), rise
from the discharge tube. The hight to which the water rises in
each tube indicates the pressure at that point. The pressure falls
off uniformly toward 0, as
indicated by the dotted
line as. The pressure of
the column a' a is required
to force the current against
the resistance of the pipe
N O. The force urging the
water from a' to h' along
the pipe is the difference
of pressure at a' and h'.
This might be called the
water motive-force between
a' and h'. If a' e' be four
times d' e', the resistance
will be four times as great
and the water-motive force
between a' and e' will be

four times as great as that between d' and e'. The fall of pressure
is the same for each unit of resistance. These facts in hydraulics
have their exact parallel in the case of electric flow, and the
analogues are too apparent to require rehearsal. The fact to be
established is that every current is due to a determinable E.M.F.
in the circuit, and fractional parts of the circuit require fractional
parts of the total E.M.F. The portion of the total E.M.F. exerted
in forcing the current through any section of the circuit is in exact
proportion to the relative resistance of such section. For example,
a battery of ten units resistance may supply an outer circuit of ten
units resistance; then one-half of the E.M.F. will be exerted
against battery resistance, and one-half against external resistance.

Fig. 399.

518

ETHER DYNAMICS.

If the same battery supply a line of 1000 units resistance, the
energy expended in the outer circuit is about ^-^% of the total
energy. The electrical efficiency of an electric generator is the
resistance of the outer circuit divided by the total resistance. In
the example above the efficiency of the battery is
1000

1000 + 10 '
485. Expenditure of eneryg.

or about

Fig. 400.

The relative resistances
of conductors carrying con-
stant currents define the
relative expenditure of en-
ergy upon such conductors.
The energy may appear as
heat, as mechanical v^ork,
or as that of chemical
decomposition. The vs^ork
done is due to the current
and to a fall of potential
along a conductor, and the
fall is determined by the
relative resistances.

This subject may be il-
lustrated graphically thus :
Let the line A B (Fig. 400)
represent the length of a
circuit (say 1000 ft.), and
the line A C the total fall
of potential ; then obvious-
ly the slope of the line
CB will represent the
average rate of fall of po-
tential throughout the cir-
cuit. But suppose that
the line for equal lengths
of the conductor varies in
resistance. Thus assume
that one-tenth the resist-
ance and consequently one-
tenth the fall of potential
(Cd) is included in the

EXPENDITURE OF ENERGY.

519

first quarter (A a), or 250 feet; then that the next 250 feet (a 6,
being very fine wire, perchance) represents one-half the total resist-
ance; the next 250 feet (be) represents one-fourth the total resist-
ance ; and the remaining resistance, fifteen-hundredths, is in the
last section (c B) of 250 feet. Then the lines C a', a' 6', h' c\ and
& B represent respectively the rat.e of fall of potential in each sec-
tion of 250 feet. The fall in each section is proportional to the
resistance, and again to this is proportional the v^ork performed by
the current in each section.

The above facts may be verified experimentally as follows : P Q
(Fig. 401) is a fine German-silver wire one meter long, stretched
along a board over a metric scale ; B is a battery ; G, a mirror
galvanometer ; K, a key for closing the circuit ; and R a coil of
German-silver wire of high resistance. When the free end of the
wire, S., is applied to any
point of the wire PQ, the
circuit is shunted through
G and R. By this means
an extremely small por-
tion of the current will be
shunted through the gal-
vanometer. This will not
perceptibly change the po-
tentials of the points P and
S. Observations are made
by placing the free end of
the wire, S, at different

points along the wire and reading the deflection produced at each
point. Kow if the potential differences are proportional to the
resistances, i.e. to the lengths of wire between the points P and S,
it follows that this resistance should have a constant ratio to the
difference of potential between the same points. But the difference
of potential is expressed by the current which it sends through the
galvanometer, so that this current (which is indicated by the deflec-
tion of the galvanometer needle) should be proportional to the
distance between P and S.

E.M.F. of a battery. The E.M.F. of a battery is considered to
be the difference of potential between its poles when the circuit is
open. If the circuit be closed, the difference of potential at its
poles will depend on the resistance of the conductor connecting
them. In this case it is

Fig. 401.

520 ETHER DYNAMICS.

in which E is the E.M.F. of the battery, R the external resistance,
and r the battery resistance. From this it follows that (1) the
difference of potential at the poles of the battery is less when the
circuit is closed than when it is open ; (2) it is less with a small
than with a large external resistance ; and (3) it is greater with a
small than with a large internal resistance.

Section XIII.

MAGNETS AND MAGNETISM.

486. Law of magnets} — Suspend by fine threads in a
horizontal position two stout darning-needles which have
been drawn in the same direction {e.g. from eye to point)
several times over the same pole of a powerful electro-magnet.
These needles, separated a few . feet from each other, take
positions parallel with each other, and both lie in a northerly
and southerly direction with the points of each turned in the
same direction.

That point in the Arctic zone of the earth towards which
magnetic needles point is called the north magnetic pole of
the earth. That end of a needle which points toward the
north magnetic pole of the earth is called the north-seeking,
marked, or -\- jpole ; this is the end that is always marked for
the purpose of distinguishing one from the other. That end
of the needle which points southward is called the south-
seeking, unmnrked, or — jmle.

Experiment 1. — Bring both points near each other ; there is a mutual
repulsion. Bring both eyes near each other ; there is a mutual repulsion.
Bring a point and an eye near each other ; there is a mutual attraction.

1 The word magnet is supposed to have been derived from the name of an ancient
district in Asia Minor called Magnesia, where was discovered at an early period a
mineral (noAV known as the magnetic oxide of iron) which possesses the property of
attracting iron. The term lodestone, or " leading stone " (inasmuch as it takes a
definite position north and south) has been given to these natural magnets.

MAGNETIC TRANSPARENCY AND INDUCTION. 521

Like poles of magnets repel, unlike poles attract each other.
487. Magnetic transparencij and induction.

Experiment 2. — Literpose a piece of glass, paper, or wood-shaving
between the two magnets. These substances are not themselves per-
ceptibly affected by the magnets, nor do they in the least affect the
attraction or repulsion between the two magnets.

Substances that are not susceptible to magnetism are said
to be magnetically transparent. When a magnet causes another
body, in contact with it or in its neighborhood, to become a
magnet, it is said to induce^ magnetism in that body. As
attraction, and never repulsion, occurs between a magnet and
an unmagnetized piece of iron or steel, it must be that the
magnetism induced in the latter is such that opposite poles
are adjacent ; that is, a N" or +pole induces a S or — pole
next itself, as shown in Fig, 402.

Fig. 402.

488. Polaritij.

Experiment 3. — Strew iron filings on a flat surface, and lay a bar
magnet on them. On raising the magnet it is found that large tufts of
filings cling to the poles, as in Fig. 403, especially to the edges ;
but the tufts diminish regularly in size from each pole towards
the center, where none are found.

Magnetic attraction is greatest at the poles, and
diminishes toiaards the center, lahere it is nothing ; i.e.
the center of the bar is neutral. This dual character
of the magnet, as exhibited at the opposite extremi-
ties of a magnet, is called polarity. If a magnet be
broken, each piece becomes a magnet with two poles
and a neutral line of its own.

489. Retentivity and resistance.
It is more difficult to magnetize steel than iron ;

on the other hand, it is difficult to demagnetize steel,

1 A word first vaguely used by Faraday in ttie sense of influence.

Fig. 403.

522

ETHER DYNAMICS.

while soft iron loses nearly all its magnetism as soon as it is
removed from the influence of the inducing body. That
quality of steel by which it resists the escape of magnetism
which it has once acquired is called its retentivity. The
greater the retentivity of a magnetizable body, the greater is
the resistance which it offers to becoming magnetized. The
harder steel is, the greater is its retentivity. Hence, highly
tempered steel is used for permanent magnets. Hardened
iron possesses considerable retentivity ; hence the cores of
electro-magnets should be made of the softest iron, that they
may acquire and part with magnetism instantaneously.

490. Forms of artificial magnets. — -Artificial magnets, in-
cluding permanent magnets and electro-magnets, are usually
made in the shape either of a straight bar or of the letter U,
according to the use to be made of them. If we wish, as in
the experiments already described, to use but a single pole,
it is desirable to have the other as far away as possible;
then, obviously, the bar magnet is most convenient. But if
the magnet is to be used for lifting or holding weights, the
U-form is far better, because the attraction of both poles
is conveniently available. Magnets, when not in use, ought
always to be protected by armatures (A, Eig. 404) of soft
iron ; for, notwithstanding the retentivity of
steel, they slowly part with their magnetism.
But when an armature is used, the opposite poles
of the magnet and armature being in contact
with each other, 'i.e. '^ with S, they serve to bind
each other's magnetism. Thin bars of steel can
be more thoroughly magnetized than thick ones.
Hence, if several thin bars (Fig. 404) be laid side
by side, with their corresponding poles turned in
the same direction and then screwed together, a
very powerful magnet is the result. This is called a com-
fo^ind magnet.

Fig. 404.

MAGNETIC LINES OF FORCE.

523

Section XIV.

MAGNETIC LINES OF FORCE. THE MAGNETIC CIRCUIT.

These lines are easily
The field of force around

491. Magnetic lines of force.
studied by the use of iron filings
a magnet is shown
by placing a paper
over it, dusting
filings upon the
paper, and tapping
it. The filings
take symmetrical
positions, form
curves between the
poles of the mag-
net or magnets,
and show that the
lines of force con-
nect the o][)j)osite
poles of the mag-
net. The fact is, that each filing, when brought within the
influence of the magnetic field, becomes a magnet by induc-
tion, and of necessity tends to take a definite position which

Fig. 405.

-^:a

'^ /

— >

:: :

S )?^>-

/ilWV^^- -■

Fig. 406.

524

ETHER DYNAMICS.

represents the resultant of the forces acting upon it from
each pole of the system. Eig. 405 represents a magnetic
field photographed from a specimen paper, and Eig. 406 is a

,/ /

^ \

--^-''

Fig. 407.

graphical representation of the same. In this illustration
the unlike poles of two magnets are placed opposite each
other. Fig. 407 is a diagram of paths of lines of force of a
bar magnet, and Fig. 408 of a horseshoe magnet.

492. Magyietic circuit. — A line of
force is assumed arbitrarily to start
from the N-pole and to pass through
the surrounding medium (e.g. the air),
entering the magnet by the S-pole,
and completing its path through the
magnet itself to its starting-point
(the IST-pole), thus forming a complete
circuit (Fig. 407). These lines do
not all emerge, however, from the
extremities. A multitude of lines
start from all parts of the magnet
and enter at corresponding points on
the other side of its central or neutral
line. No magnetic line of force can exist without completing

y / I

Fig. 408.

LINE OF FORCE THE AXIS OF ETHER WHIRLS. 525

its own circuit/ and lines of force never cross or merge into
one another, consequently a magnet cannot have a single pole.
Lines of force possess several peculiar characteristics. One
is that in air and most other mediums they tend to separate
from one another, but at the same time tend to become as
short as possible. The strain is as if these lines were
stretched elastic threads endowed with the property of repel-
ling one another as well as of shortening themselves ; in
other words, there is tension along the lines and pressure at
right angles to them. Air is a poor conductor for lines of
force, or its 'permieability'^ is low ; on the other hand, iron
has high permeability for lines of force, and if a
piece of iron be brought within a magnetic field, a [

portion of the lines of force will crowd together
into it, leaving their normal paths through the air
for a medium of greater permeability.

493. Line of force the axis of ether luhirls,

A line of force is supposed to represent the axis of
a series of ether whirls. Fig. 409 gives a crude pic-
torial representation of the supposed constitution of
the whirls of an electro-magnetic line of force. A
series of curtain rings might be strung
upon a stretched thread and caused to
rotate around it. This would give some
idea of the hypothesis ; the thread
would give the direction of the line of
force and its conventional representa-
tion as a simple line. The perfectly cir-
cular line of force is such as those (§ 445) i
surrounding a wire carrying a current, 1
and represented in Fig. 410. Its mechanical analogue ^
is seen in a smoke-rino^ such as is sometimes caused to Fig. 409.

Fig. 410.

1 Herein exists a notable difference between these and electrostatic lines of force.
Every electrostatic charge is bound, i.e. has an opposite and equal charge somewhere.
To this its lines of force go ; but there is no circuit, there is only a connection.

2 Permeability is a quality christened by Lord Kelvin.

626

ETHER DYNAMICS.

rise from the bowl of a tobacco pipe by skillful operation. It is
well illustrated by the rings arising from the spontaneous combus-
tion of phosphureted hydrogen in air. These show the whirling,
rotary motion around the circular axis of the ring. Such are called
vortex rings.

494. Attraction and rejndsion between magnetic poles.

The tendency of these whirls is to bulge out by reason of centri-
fugal force. An assemblage of such parallel whirls would compress
each other laterally and cause a longitudinal tension (§ 425). On
this hypothesis the phenomena of magnetic attraction and repulsion
are explainable.

If the N-pole of one magnet be placed opposite the S-pole
of another (Fig. 406), the lines of force issuing from the
former will enter the latter, and, tending to shorten, will
produce attraction. If the similar ends be opposed (Fig. 411),

Fig. 411.

the lines of force will be turned away from each pole in all
directions, and will complete their circuits independently.
Thus becoming parallel they will repel one another; for
this reason like magnetic poles repel each other.
495. Equipotential surfaces.

The potential at any point in a magnetic field is the quantity of
work that would have to be done in bringing a + unit of magnetism
from an infinite distance to that point.

If a magnetized particle be moved either toward or from a mag-
netic pole, work is done either by or against the attracting or re-
pelling force, and the particle is said to be moved from a point
where the magnetic potential has one value to a point where it has

TUBES OF FORCE.

527

another value. Imagine an N-pole isolated from its companion
S-pole. We may suppose that all points at equal distances from
this pole are at the same potential, and these points joined form a
spherical surface. The potential at every point of this surface
(being the same) may be represented by V. Within and without
this lie successive spherical equipotential surfaces over each of which
the potential is constant, but the potential at each surface differs
from that at any of the others. No work is required to move a
quantity of magnetism from one point on an equipotential surface
to another point on the same surface. Equipotential surfaces are
not necessarily spherical. But whatever be their form, the lines of
force are always at right angles to the equipotential surfaces.

496. Tubes of force.

Suppose A B (Fig. 412) to be a section of an equipotential surface.
Lines of force pass through the surface, those grazing its edges cut-
ting off an area. A' B', from another equipotential surface. The
space comprised between these equi-
potential areas and bounded laterally
by lines of force is called a tuhe of
force. This space may be supposed
to be filled with a bundle of lines of
force. Now the intensity of the
magnetic force at the potential A' B',
as compared with that at A B, is in
inverse proportion to the magnitudes
of their respective areas.

The intensity of magnetic forces
at any two points may be compared
by stating the relative numbers of
the lines of force which pass through
equal units of area of the equipoten-
tial surfaces containing the points

compared. The fewer these lines per unit area, the less the local
intensity of the force. The strength of field between the poles of a
magnet may be expressed in dynes, but now it is more common
among electricians to express this strength by the number of lines
of force per cm^, each line of force representing a dyne.

For example, if the strength of the magnetic field, or the force
on a unit pole, be ten dynes at any point, then ten lines of force are

Fig. 412

528

ETHER DYNAMICS.

said to pass through an area of 1 cm^ held perpendicularly to these
lines.

497. Strength of magnetic field by the method of oscAllatioiis.

This method of comparing the strength of the magnetic field at
different points in it depends on the principle that a pendulum
makes isochronous oscillations when the length of the arc is very
small, and that the force which is always pulling it back to its
position of rest is proportional to the square of the number of oscil-
lations in a given time.

A magnetic needle about 3 in. long is suspended by a silk fiber
in the magnetic meridian. The needle is deflected from the merid-
ian and allowed to vibrate, and the number of vibrations made in
a given time are counted. Suppose that 11 oscillations be made
in 30 seconds under the action of the earth. If E represent the
strength of the earth's magnetic field, then E may be measured by
112= 121.

Then on placing a long bar magnet (Fig. 413) in a vertical position
in the same meridian with the needle, and its N-pole opposite the
S-pole of the needle, the two poles being 4 cm apart,
suppose that the number of oscillations made in 30
seconds be 61. If, then, M4 represent the intensity
[ of the field at a distance of 4 cm, E + M4 is measured
by 512 = 2601. Hence M4 is measured by 2601 — 121
= 2480.

Suppose that on removing the pole to a distance
of 8 cm the number of oscillations is 27. Denoting
by Ms the intensity of the field of the magnet at a
distance of 8 cm, we have E + Ms measured by 272 =
729, and therefore Ms is measured by 729 — 121 =
608. Hence M4 : Ms = 2480 : 608 = 4 : 1 (within the
Fig. 413. limit of errors of observation).

498. Latu of inverse squares. — In this manner may be
demonstrated experimentally the Law of Inverse Squares as
applied to magnetism, viz. : The force exerted between two
magnetic poles varies inversely as the square of the distance
between them.

The magnitude of the force in any case is numerically equal

DEFINITIONS. 529

to the product of the strength of the poles divided by the square
of the distcuice between them;

7n7}i'

^~~d^'

The C.G.S. unit pole is that pole ivhich repels an equal pole
placed a centimeter away ivith a force of one dyne}

499. Definitions. — We are now in a position to understand
the following definitions : A portion of space throughout which
magnetic effects are exerted by a distribution of magnetism is
called a magnetic field. At each point of the field a pole of
intensity, m, is acted on by a definite force. . The intensity
of a magnetic field at a given point is equal to the force in
dynes with which a unit pole would be acted upon at that
point. Let H denote the intensity of field at any point ;
then the force actually exerted at that point on a pole of
strength m is m H.

A line of magnetic force is a line drawn in such a manner
that the tangent to it at each point is in the direction of the
resultant magnetic force at that point.

The magnetic potential at any point is the work that must
be done against the magnetic forces in bringing up a unit
magnetic pole from a point at an infinite distance to the
given point. The difference of magnetic ptotential between
two points in the field is the work done in transferring a unit
magnetic pole from one of these points to the other, against
magnetic forces. A surface in which the magnetic potential
at all its points is the same is an equipotential surface.

Let m be the quantity of magnetism at one of the poles of
a magnet, and I the distance between the poles ^ ; the magnetic
moment of the magnet is the product m I.

1 In practice it is impossible to obtain a single isolated pole ; the total quantity of
(+ and. — ) magnetism in any magnet is, algebraically reckoned, zero.

2 In a steel bar magnet the poles are not strictly at tbe extremities, and hence
tbe magnetic length is a little less tban tbe actual length of the magnet. For most
bar magnets, the magnetic length is about .83 of the actual length. The magnetic
length of a horse-shoe magnet is the shortest distance between its poles.

530

ETHER DYNAMICS.

The intensity of mag7ietization is the ratio of the magnetic
moment of a magnet to its volume.

A magnetizable body placed in a magnetic field becomes
magnetized in the direction of the lines of force of the field.
Its magnetism is called induced magnetism, and the action
itself is called rnagnetic induction. The magnetism retained
by a magnetic body after it has been withdrawn from the
field is called residual magnetism.

Section XV.

TERRESTRIAL MAGNETISM.

500. The earth a magnet.

A dipping-needle is so supported that it can revolve in a vertical
plane. Indifferent equilibrium is first established in the steel
needle, so that if placed in a horizontal (or any other) position it
will rest in that position. Then it is strongly magnetized. After-
ward it will take the horizontal position only at the magnetic
equator of the earth.

Experiment 1. — Place a dipping-needle over the + pole of a bar
magnet (Fig. 414). The needle takes a vertical position with its — pole
down. Slide the supporting stand along the bar ; the — pole gradually

rises until the stand reaches
Nra K, - c ^s the middle of the bar,

where the needle becomes
horizontal. Continue mov-
ing the stand toward the
S — pole of the bar ; after
passing the middle of the
bar the + pole begins to
dip, and the dip increases until the needle reaches the end of the bar,
when the needle is again vertical with its + |)ole down.

If the same needle be carried northward or southward along the
earth's surface, it will dip in the same way as it approaches the polar
regions, and be horizontal only at or near the equator.

Fig. 414.

MAGNETIC POLES OF THE EAKTH.

531

Experiment 2. — Support a small pane of window glass on a table, by
placing under the glass near its corners four slices of cork about one-
eighth of an inch thick. Be-
neath the center of the glass
on the table place a chcular
disk of magnetized steel. Sift
iron turnings upon the upper
face of the glass through a
fine wire sieve. Gently tap
the glass at convenient points
with the end of a lead-pencil.
The filuigs arrange themselves
in Imes radiating from each
pole.

Experiment 3. — Suspend
a small magnetized cambric
needle by a fine thread at its
center and carry it around
the disk (Fig. 415). The
needle passes through all the
phases stated in Exp. 1, so
that we may fancy the disk

to be the earth, and study therefrom, in a general way, the changes
that the needle undergoes, as it is carried around the earth in a norther-
ly or southerly direction.

Fig. 415.

The last experiment presents a true exhibition^ on a small
scale, of what the earth does on a large one, and thereby
presents one of many phenomena which- lead to the conclu-
sion that the earth is a magnet. In other words, these
phenomena are just what we should expect if a huge magnet
were thrust through the axis of rotation of the earth, as
represented in Fig. 416, — having its jST-pole near the S geo-
graphical pole, and its S-pole near the N geographical pole \
or if the earth itself were a magnet.

501. Magnetic poles of the earth. — It will be seen that
there are two points where the needle points directly to the
center of the disk. A point was found on the western coast

532

ETHER DYNAMICS.

of Boothia, by Sir James Ross, in the year 1831, where the
dipping-needle lacked only one-sixtieth of a degree of pointing

directly to the
earth's center. The
same voyager sub-
sequently reached
a point in Victoria
Land where the
opposite pole of
the needle lacked
only 1° 20' of
pointing to the
earth's center.

It will be seen
that, if we call that
end of a magnetic
needle which
points north the
N-pole, we must
call that magnetic pole of the earth which is in the northern
hemisphere the S-pole, and vice versa. (See Fig. 416.) Hence,
to avoid confusion, many careful writers abstain from the use
of the terms north and south poles, and substitute for them the
tQYm^ positive and negative, or marked and unmarked poles.

502. Variation of the needle. — Inasmuch as the magnetic
poles of the earth do not coincide with the geographical poles,
it follows that the needle does not in most places point due
north and south. The angle which the vertical plane through
the axis of a freely suspended needle makes with the
geographical meridian of the place is known as the angle of
declination. In other words the angle of declination is the
angle formed by the magnetic and geographical meridians.
This angle differs at different places. The magnetic axis of
a needle is a straight line connecting its two poles.

Fig. 416.

ISOGONIC CURVES.

533

Experiment 4. — We can easily find, as did Columbus, the declination
at any place by the following method : Set up two sticks so that a string
joining them will lie in the same vertical plane with the Pole Star ; the
string will lie in the geographical meridian. Place a long magnetic
needle over the string ; the angle between the needle and the string is the
required declination. K great accuracy be required, allowance must be
made for the fact that the star is not exactly over the pole, but appears
to describe daily around it a circle whose diameter is at the present time
about 2i°.

503. Isotonic curves. — These are lines connecting all points

Fig. 417.

of equal declination on the earth's surface. The line of no
declination or isogonic of 0° (Fig. 417) commences at the N.
magnetic pole about lat. 70°, long. 96°, passes in a south-
easterly direction across Lake Erie and Western Pennsyl-
vania, and enters the Atlantic Ocean near the boundary
between the Carolinas. Pursuing its course through the
south polar regions, it reappears in the eastern hemisphere
and crosses Western Australia, the Caspian Sea, and thence
to the Arctic Ocean. There is also a detached line of no
declination inclosing an oval area in Eastern Asia and the
Pacific Ocean. In the eastern (or Atlantic) hemisphere,

534

ETHER DYNAMICS.

bounded by the line of no decimation, the declination is
westward, as indicated by continuous • lines in the figure.
In the western (or Pacific) hemisphere the declination is
eastward, as indicated by dotted lines.

The magnetic poles are not fixed objects that can be located
like an island or cape, but are constantly changing. They
appear to swing, something like a pendulum, in an easterly
and westerly direction, each swing requiring centuries to
complete it. The north magnetic pole is now on its westerly
swing.

504. Inclination or dip. — We have seen that in the northern
hemisphere the lines of force tend downward and northward.

A magnetic needle thus
tends to place itself so
that its axis points down-
ward and to the magnetic
pole of the earth. The
angle which the axis of
a freely suspended mag-
netic needle makes with
the horizontal plane is
called the inclination or
dip of the needle. Fig.
418 represents a dipping-
needle such as is used
in determining magnetic
inclination, and Fig. 419
represents a declinometer
for determining the decli-
nation. A is a mounted
telescope for sighting
north star. Its axis is

Fig. 418.

some astronomical object, e.g. the
levelled by the spirit level B.

The line passing in an easterly and westerly direction

INTENSITY.

535

Fig. 419.

around the earth along which the lines of force (or needle)
are horizontal, i.e. at which
the dip is zero, is the magnetic
equator. It does not coincide
with the geographical equator.
The lines roughly parallel to
the magnetic equator, along
Avhich the dip is equal, are
the magnetic parallels. These
are lines along which equipo-
tential surfaces cut the sur-
face of the earth.

We have before noted the
fact that lines of force are
always at right angles to equi-
potential surfaces, and conse-
quently at the magnetic poles
where the dip is 90° the equipotential surfaces are tangent
to the earth's surface.

505. Intensity. — The force, expressed in absolute measure,
with which the earth's magnetism urges a unit magnetic pole
(§ 498) is the intensity of the earth's magnetic field ^ at the
place.

The earth's action on a needle is a mechanical couple,
the effect of which is to cause only a rotary motion. This
is what is meant when the earth's action on the needle is
said to be directive only.

506. Connection betiveen the sun and the earth^s magnetism.

1 If the inclination be found, and the horizontal component of the intensity of
the earth's field acting upon a unit pole be known, we have the data reqiiired for
determining by parallelogram of forces the whole intensity of the earth's magnetic
field in the direction of the lines of force at any point. Such a measurement is of
importance in refined work, since it consists essentially in determining the couple
which must be exerted by the earth's magnetic force on a needle in order to balance
that produced by the current. For methods of determining these quantities, see
Cumming's Electricity, or any complete laboratory manual.

536

ETHER DYNAMICS.

Magnetic storms, or disturbances of the earth's magnetism
coincident with outbreaks of sun spots and solar storms, point
to an undoubted connection or sympathy between the sun and
the earth's magnetism ; but of the nature of this connection
our knowledge is as yet very limited.

Section XVI.

MAGNETIC RELATIONS OF THE CURRENT. ELECTRO-MAGNETS.

507. Magnetic field due to a circular current. — If a wire
be bent into the form of a circle of about 10 in. diameter,
and placed vertically in the magnetic meridian, and a card-
board be placed at right angles to the circle so that its hori-
zontal diameter is coincident with the upper surface of the
cardboard, and a very strong current be sent through the
wire in the direction indicated by the arrow-head in the wire,
iron filings sifted upon the card will arrange themselves as
shown in Fig. 420. And if a freely suspended test-needle

Fig. 420.

be carried inside and outside the circle, the several positions
taken by the needle, as indicated in the figure by arrows,
corroborate the directions of the lines of force as indicated
by the filings.

MAGNETIC CURRENT.

537

If the direction of the current be reversed, the direction of
the needle will be reversed wherever it may be placed.

The direction indicated by the is'-pole of the magnetic
needle placed anywhere in the field is called the positive
direction of the lines of force.

508. Magnetic current. — In fact, when a current traverses
a wire (or other conductor) lines of force encircle the electric
current at right angles to it.^ The electric current and its
encircling lines of force always co-exist, and one varies directly
as the other when there is no magnetic substance near the
wire. The direction of the encircling lines of force with
reference to the electric current may be illustrated by the
use of a corkscrew. The direction of the electric current
corresponds to the propul-
sion of the point of the
corkscrew when entering
a cork, and the direction
in which the screw is
turned or the hand is
twisted in propelling it
corresponds to the direc-
tion of the lines of force.

If a circle of wire bear-
ing a very strong current
be freely suspended and a ^^^- ^'^'^•

pole of a very strong bar magnet be presented to one of its

1 " Every conducting wire is surrounded by a sort of magnetic whirl. A great part
of the energy of the so-called electric current in the wire consists in these external
.magnetic whirls. To set them up requires an expenditure of energy ; and to main-
tain them requires a constant expenditure of energy. It is these magnetic whirls
which act on magnets, and cause them to set, as galvanometer needles do, at right
angles to the conducting wire." — S. P. Thompson.

It may be demonstrated that a law analogous to Ohm's Law Is applicable to the
magnetic whirl or "flux." Let C=:the strength of the magnetic flux (or field)
expressed in lines of magnetic force ; M = the force which gives rise to the magnetic
flux, called the magneto^notive force; and 11=: the resistance to the magnetic flux ;

thenC=rM.
K

538

ETHER DYNAMICS.

faces, the circle will be attracted or repelled according to "the
nature of the pole and the direction of the current. We may
consider a circular current as if it were a very short magnet,
one face of the circle being the north end, and the other face
the south end, as represented in Fig. 421.

Observe (Fig. 420) that the directions of the filings near
the center of the circle lie nearly parallel with its axis, but
outside to the right and left of the axis the filings lie in
curves around each wire.

509. Solenoid. — An insulated wire wound into a helix of
considerable length as compared with its diameter is called

a solenoid} It is evident
that the intensity of the
magnetic field must be
greatly increased by the
joint action of the many
current turns. The mag-
netic field within the sole-
noid is nearly uniform in
strength, and the lines of force to within a short distance of
its ends are parallel with its axis, as shown in Fig. 422.^

510. Magnetic polarity of electro-magnetic solenoid. — Fig.
423 represents a small battery floating on water. The lead-
ing wire of the cell is
wound into a horizon- (#oouoouo^ ,^
tal solenoid. Slowly
after the cell is floated
it will take a position
so that the axis of
the solenoid will point
north and south like fj^. 423.

Fig. 422.

1 Faraday first applied the term solenoid to a system of circular currents parallel
with one another.

2 An open solenoid of a single layer is here given, in order the better to show the
directions of the several lines of force.

POLARITY OF ELECTRO-MAGNETIC SOLENOID. 539

2 I llllllMHIliP

Fig. 424.

a magnetic ceedle. Hold (say) the S-pole of a bar magnet
near that end of the solenoid which points north; the solenoid
is attracted by the magnet. Hold the N-pole of the magnet
near the north-pointing end of the solenoid; the magnet
repels the solenoid.

Eepeat the above, using in place of the bar magnet another
solenoid (Fig. 424) ; there will
be a repetition of the same phe-
nomena as obtained with the bar
magnet. Introduce a rod of soft
iron into the solenoid held in the
hand, thereby making of it an
electro-magnet; the only change
observed ■ is that the force of
attraction and repulsion is greatly increased.

Place the wire of another battery over and parallel with
the coil (Fig. 425), so that the two currents will flow in planes

at right angles with
^y >^? — ^ — ^^'^ each other. The

coil is deflected like
a magnetic needle
(Fig. 426).

Eeverse the di-
rection of the cur-
rent above and the
deflection is reversed.

We thus prove that a solenoid bearing a current possesses
polarity as if it were a magnet, and that there can be pro-
duced by a current-bearing solenoid a magnetic field of the
same character as that produced by a permanent magnet.
There is no essential difference between a permanent magnet,
a current-bearing solenoid, and an electro-magnet, except that
the last may be made much stronger than either of the
others.

Fig. 425.

Fig. 426.

540

ETHER DYNAMICS.

Given the direction of the current in a solenoid, to find the

N~ and S-poles of the solenoid, and vice versa.

Rule 1. Place the palm of the right hand against the side

of the solenoid so that the fingers will point in the direction

of the current passing
through the windings
(as shown in Eig. 427) ;
the thumb will point in
the direction of the N-
pole of the solenoid or
electro-magnet}

EuLE 2. Ascertain
the N-pole of the sole-
noid or electro-magnet
with a Tnagnetic needle,

and place the palm of the right hand upon the solenoid so that

the outstretched thumb points in the direction of the N-pole ;

the fingers will point in the di7Xction in which the current

passes in the windings.

DIRECTION OF
MAGNETIC FORCE

Fig. 427.

Section XVII.

ELECTRODYNAMICS. AMPERE S THEORY OF MAGNETISM.

511. MutiLal action of currents on one another. — We have
hitherto discussed the direction of the magnetic fiekl due to
a straight current, have determined the direction which a
test-needle takes in virtue of the action of the current field,
and have learned to regard an electric current as producing
north and south polarity along the whole circuit of the cur-
rent-bearing wire. That is, if we suppose that a test-needle
be moved up or down just back of the current-bearing wires

1 The following suggestion will often prove of practical value : that is the south
pole of a helix where the current corresponds to the motion of the hands of a watch,
S^and that is the north pole where the current is in the reverse direction, *N.

MUTUAL ACTION OF CURRENTS.

541

(Fig. 428), the N- and S-poles will take the positions indi-
cated by n and s. We may readily premise from inspection
of the polarity developed, that if the wires were so sus-
pended as to be free to move either toward or from each
other, the pair of wires in which the currents flow parallel to
each other and in the same direction. A, would attract each
other, and the pair of wires in which the currents flow in
opposite directions, B, would repel each other ; but if the
currents be inclined to each other as in Fig. 429, they will

t t

t i

A B

Pig. 428.

Fig. 429.

tend to move into a position in which they will be parallel
and in the same direction. That such actually takes place
may be shown by the following experiments : —

Experiment 1. — Eig. 430 represents a portion of a divided circuit.
The lower ends of the wires dip about one-sixteenth of an inch into
mercury, and the wires are so suspended that they are free to move to-
ward or from each other. Send a current of a battery of three or four
Bunsen cells, in multiple arc, through this divided circuit. The two
portions of the current travel in the same direction and parallel with each
other, and the two Avires at the lower extremities move toward each
other, showing an attraction.

Experiment 2. — Make the connections (Fig. 431) so that the current
will go down one wire and up the other. They repel each other.

542

ETHER DYNAMICS.

Experiment 3. — Send a current through the spiral wire represented
in Fig. 432. Here the current flows nearly parallel with itself, and the
attraction causes the coil to contract and to be lifted out of the cup of
mercury below. But the instant it leaves the mercury the circuit is
broken, the current and attraction cease, and the wire dips into the
mercury again. Thus rapid vibratory motion of the coil is produced.

In the experiment with the floating cell and current-bearing
wire placed over and parallel to the solenoid (Fig. 425), a
careful examination will disclose the fact that not only do
the planes in which the current flows in the coil tend to

ff ii t ^j^^ ' "^^

Fig. 430.

Fig. 431.

Fig. 432.

become parallel to the current above, but that the current in
the upper half of the coil, where the influence due to prox-
imity is greatest, tends to place itself so as to flow in the
same direction as that of the current above.

512. Ampere's Laws. — Law 1. Parallel currents, if in the
same direction, attract one another ; and if in opposite direc-
tions, they repel one another.

Law 2. Currents that are not parallel tend to become parallel
and floiD in the same direction.

A little reflection will show that the observed motion is
the expression of a tendency on the part of any movable
current to cut lines of magnetic force at right angles, the direc-

ampere's theory of- magnetism. 543

tion of motion being reversed when the direction either of
the lines of force or of the current is reversed. This prin-
ciple is of immense importance from an industrial aspect.
The most important outcome of its application is the dynamo-
motor (§ 533) by means of which electrical energy is converted
into TYieclianical energy, and through the agency of which
electric street cars are j)ropelled.

513. Am,jpere^s theory of magnetism. — This celebrated theory
briefly stated is that magnets and solenoid systems are funda-
mentally the same; that magnetism is simply electricity in
rotation, and that a magnetic field is a sort of whirlpool of
electricity. Not, of course, that a steel magnet contains an
electric current circulating round and round it as does an
electro-magnet, but that every molecule of iron, steel, or
other magnetizable substance is the seat of a separate current
circulating round it continuously and without resistance, and
thus every molecule is a magnet.

According to the theory, in an unmagnetized bar these
currents lie in all possible planes, and, having no unity of
direction, they neutralize one another, and so their effect as
a system is zero. But if a current of electricity or a magnet
be brought near, the effect of the induction is to turn the
currents into parallel planes, and in the same direction, in
conformity to Ampere's Second Law. If the retentivity be
strong enough, this parallelism will be maintained after the
removal of the inducing cause, and a permanent magnet is
the result.

Intensity of magnetization depends on the degree of paral-
lelism, and the latter depends on the strength of the influ-
encing magnet. When these currents have become quite
parallel, the body has received all the magnetism that it is
capable of receiving, and is said to be saturated. Although
the currents really circulate around the individual molecules,
yet tlie resultant of these forces is essentially the same as if

544

ETHER DYNAMICS.

a superficial sheet of currents circulated around the body as a

whole. Fig. 433 represents sections of a cylindrical magnet, and

the included circles represent
the circulation of the several
currents around the mole-
cules lying in these sections.
It will be seen that the cur-
rents at the contiguous sides
of any two of these circles
move in opposite directions,
and therefore must neutral-
^ ize each other; while the

^^^' *^^- currents that pass next the

circumference of the magnet are not so affected.

514. Rotation of a ruagnetic pole round a current, and of

a current round a magnetic pole. — A current and a magnetic

pole neither attract nor repel

each other, but tend to rotate

about each other, the action

being at right angles to the line

joining them. Hence a mag-
netic pole free to move will

rotate round a current. This

may be shown experimentally

with apparatus like that shown

in Fig. 434. The magnet NS

is bent so that it may be pivoted

on its middle point, the current

being brought to an annular

mercury cup, A, by means of a

wire which dips into the mer-
cury, leaving through the screw

cup B. When a strong current is passing, the magnetic pole

N rotates steadily, and by reversing the direction of the

current the direction of rotation of the pole is reversed.

Fm. 434.

ROTATION OF A MAGNETIC POLE.

545

The rotation of a current round a pole may be shown by
pivoting a wire bent in the form of an inverted letter U A
(Fig. 435), on the top of a vertical
horse-shoe magnet. The divided current
passes through the mercury cups B and
C, and leaves by the annular cups D and
E which surround the magnet lower
down. The cups B and C and the wires
passing from them to the cups D and E
are so pivoted upon the extremities of
the magnet as to be free to rotate around
its poles. When a strong current is
passed through the circuit the wires will
rotate in opposite directions round its
two poles.

The hypothetical currents that circulate round a magnetic
molecule we shall call amperian currents, to distinguish them
from the known current that traverses the solenoid. In
strict accordance with this theory, the poles of the electro-
magnet are determined by the direction of the current ^in the
helix. The inductive influence of the electric current causes
the amperian currents to take the same direction with itself,
as represented in Fig. 436.

By the amperian theory the earth's polarity is accounted

Fig. 435.

Fig. .436.

for by assuming it to be girdled by electric currents, called
earth currents, in planes approximately parallel to the equa-
tor. A. person standing on the Arctic magnetic pgle of the
earth would, if the currents were visible to him, see them

546

ETHER DYNAMICS.

(more properly their resultant, a current sheet) circulating
round him towards his right from east to west, or in the same
direction as the sun appears to him to go round the earth.
According to Ampere's theory it is the tendency of the nearer
portions of the earth currents and the amperian currents circu-
lating round a magnetic needle to coincide in direction and to
be parallel ; that causes the needle to point north and south.
However well adapted this theory may be to explain most
of the known phenomena of magnetism, it should be borne in
mind that physicists of this generation value the theory
rather as a help to the imagination and memory, than as a
true statement of the facts. It is nearer the truth to say that
the molecules are polarized as if currents were circulating
around them ; of the actual existence of such currents we
know nothing. So also of the real nature of polarity we
know little or nothing.

Section XVIII.

ELECTKO-MAGNETIC INDUCTION.

515. Description of apparatus.

Fig. 437.

— A (Fig. 437) is a short coil
of coarsB wire {i.e. the wire
which it contains is com-
paratively short), and has,
of course, little resistance.
B is along coil of fine wire
having many turns. Coil
A is in circuit with two
Bunsen cells in multiple arc.
This circuit we call the
primary circuit, the current
in this circuit the primary
or inducing current, and the
coil the primary coil. An-

DESCRIPTION OF APPARATUS.

547

other circuit, having in it no battery or other means of gener-
ating a current, contains coil B and a galvanoscope with an
astatic needle.^ This circuit is called the secondary circuit,
the coil the secondary coil, and the currents which circulate
through this circuit are called secondary or induced currents.

Experiment 1. — After all the connections are made, and a current is
established in the primary circuit, and the galvanoscope needle is brought
to zero, lower the primary coil quickly into the secondary coil, watching
at the same time the needle of the galvanoscope to see whether it moves,
and, if so, in what direction. Simultaneously with this movement there

Fig. 438.

is a movement of the needle, showing that a current must have passed
through the secondary circuit. Let the primary coil rest within the
secondary, until the needle comes to rest. After a few vibrations the
needle settles at zero, showing that the secondary current was a tem-
porary one. Now, watching the needle, quickly pull the primary coil
out ; another deflection in the opposite direction occurs, showing that a
current in the opposite direction is caused by withdrawing the coil.

It is evident that in this case the current does not by its
mere presence cause an induced current, but that a change in
the relative positions of the two circuits, one of which bears
a current, is necessary.

1 This needle consists of two needles of about the same intensity with their poles
reversed, fixed parallel with each other. Though the needles nearly neutralize each
other and are therefore little affected by the field of the earth's magnetism, they are
especially sensitive to the influence of the electric current properly situated.

548 ETHER DYNAMICS.

Experiment 2. — Place tlie primary coil within the secondary. Open
the primary wire at some point and then close the circuit (Fig. 438) by
bringing in contact the extremities of the wires. A deflection is pro-
duced. As soon as the needle becomes quiet, break the circuit by sepa-
rating the wires ; a deflection in the opposite direction occurs.

The same phenomena occur when the primary current is
by any means suddenly strengthened or weakened.

An examination of the direction of tliese currents enables
us to state the facts as follows : Starting a current in a
primary, increasing the strength of the primary current, or
moving the primary nearer while the current is steady, pro-
duces a transitory current in the opposite direction in the
secondary. Stopping the primary, diminishing the strength
of the primary current, or moving the primary away while
the current is kept steady, causes a transitory current in the
same direction in the secondary.

It is evident, therefore, that the conditions under which a
current in the primary coil can cause a current in a neighbor-
ing secondary depend upon some change either in the strength
of the primary current or in the relative positions of the
primary and secondary circuits.

Experiment 3. — Introduce the bundle, D (Fig. 437), of soft iron wires,
called the core^ into the primary coil, and make and break. the primary
circuit as before. The deflections are now very much increased.

Experiment 4. — Substitute a person for the galvanometer in the sec-
ondary circuit, the person grasping some metallic handles made for the
purpose and used as electrodes. The person experiences at the instant
of making and breaking a peculiar sensation in his wrists and arms,
called a shock.

Experiment 5. — Introduce into the primary circuit the automatic
make-and-break piece C (Fig. 437). Remove the core from the primary
coil. Let a person grasp the electrodes of the secondary circuit. This
person experiences a series of shocks which seem to him almost, if not
quite, continuous. These shocks can be intensified to suit the pleasure
of the person who is receiving them, by gradually lowering the core into
the x^rimary coil.

LAW GOVEKNING E.M.F. OF INDUCED CURRENTS. 549

Experiment 6. — Eeflecting that you have hitherto found a coil of
wire having a current passing through it acting as a magnet, you have
now an opportunity to try the
converse, i.e. to see whether a
magnet may not take the place of
a current-hearing coil. Introduce
suddenly a bar-magnet (Fig. 439)
into the secondary coil, as in Ex-
periment 1 ; a deflection is pro-
duced. "Withdraw it and an op-
posite deflection occurs.

The act by wliich the prim-
avj, or a magnet, causes a cur-
rent in a neighboring second-
ary is called inagneto-electric
induction.

516. Law governing E.M.F. of induced currents. — In any
induced current the E.M.F. at any instant is jproportional to
the rate of change in the number of lines of force passing through
the circuit at that instant. If there be no change in the
number of lines there is no induced E.M.F., however rapid
the motion may be.

Experiment 7. — Introduce a long bar magnet, NS (Fig. 440), into a
short coil of wire, C, connected to a galvanometer, G. Place the coil

half-way between the poles of
c

Fig. 439.

Fig. 440.

the magnet *and move it rapidly
(say) one centimeter toward either
pole ; no movement of the gal-
vanometer needle occurs although
all the lines of force of the mag-
net pass through C.

Now place C near one of the
ends of the magnet ; a similar mo-
tion produces a large deflection.

By means of an induction coil, a current of a few amperes
circulating in the primary under an E.M.F. of not more than

550

ETHER DYNAMICS.

10 or 20 volts can be caused to yield currents in the secondary
urged by an E.M.F. of many thousand volts.

517. Faraday^ s law of induction. — If any conducting cir-
cuit be placed in the magnetic field, then, if a change of
relative position or change of strength of the primary current
cause a change in the number of lines of force passing through
the secondary, an electro-motive force is set up in the sec-
ondary proportional to the rate at which the number of lines
of force included by the secondary is varying.

Consider the case of induction by a magnet. Let S (Fig.
441) be a secondary circuit and IST a magnet projecting a cer-
tain number of lines of force
through the circuit. If S
be moved nearer to the mag-
net, say to S', a'much greater
number of lines of force of
the magnet pass through the
circuit than when in its for-
mer position, owing to the
divergence of the lines as
they recede from the pole.
We may now understand,
in part, the reason why a core of soft iron so greatly increases
the induced current. It acts like a lens in focusing or con-
centrating more lines of force from the magnetic inducer
through the aperture of the secondary, and therefore any
movement makes a greater rate of change, and hence a greater
induced electro-motive force.

518. Tlarth induction. — Call to mind that the earth itself
is a great magnet, and that its lines of force pass through
our atmosphere from pole to pole, and it will be easy to con-
ceive that the mere motion of a coil of wire about an axis
properly placed ^ is all that is necessary to produce a current.

Fig. 441.

1 The coil should he placed at right angle
locality. Why ?

to the direction of the dip at, the

EARTH INDUCTION.

551

Such a coil with a galvanometer G in circuit is represented
in Fig. 442. The rotation of the coil across the magnetic

Fig. 442.

lines of force of the earth effects a change of the number of
lines of force passing through it, as may be understood by
inspection of Fig. 443, and this creates temporary currents
in the coil.

By examination of Fig. 443 it will be seen that there are
in each complete rotation of the coil two points (as A in the
figure) where the coil encloses
a maximum number of lines of
force. In this position the in-
duced current vanishes, for at
this instant the number of
lines is neither increasing nor
diminishing. As the coil in
its rotation approaches these
points, the number of lines of
force increases, and after leav- fig. 443.

552 ETHER DYNAMICS.

ing it the number diminishes. This will evidently cause a
change in the direction of the induced current twice during
each revolution. If then by means of a covimutator, a (Fig.
452), the direction of the current in the galvanometer be
changed relatively to its direction in the coil at each half
revolution, we have an intermittent current constant in direc-
tion through the galvanometer.

519. Lenz^s law. — Recurring to the primary and secondary
circuits we remark that the motion of the one or the other
may be in arcs of circles or in any way, yet the motion may
always be resolved so as to give a resultant indicating ap-
proach or recession. The law by which the direction of the
induced current is determined is known as Lenz's law, and
may be expressed as follows: "7?^ all cases of induction the
direction of the induced current is such as to oppose the motion
which produces itJ^ Thus approach develops an opposite cur-
rent, since opposite currents resist approach, while recession
develops a current of similar direction, since similarly directed
currents attract one another and thus resist recession.

520.. Mechanical energy transformed into electric energy^ and
vice versa. — It is, then, apparent that the current developed
in the secondary circuit is at the expense of mechanical
energy, and thus mechanical energy is transformed into
electric energy.^

Eeturning to the apparatus (Figs. 434 and 435) in which
we have the movement of a magnet-pole in the field of a
current, and of a current in the field of a magnet, — if we
replace the battery by a sensitive galvanometer, it is evident
from the above discussion that on rotating the magnetic pole
(Fig. 444) or the conductor (Fig. 445), in other words on

1 The student might have been able to prophesy Lenz's law by reasoning thus :
Suppose coil B approaches circuit A, we know (1) that electrical energy appears in
B ; therefore from the doctrine of conservation of energy we know (2) that work
must have been done ; hence, if work has been done, there must have been a repel-
lent force between A and B.

SELF-INDUCTION.

553

reversing the operations indicated in § 514, currents will
traverse the circuits and their presence may be detected by
the deflection of the needles of the galvanometers G. If we
examine the direction of
these induced currents,
we shall discover that
they are always opposite
to the current which
would actually cause the
rotation. Since the in-
duced current is opposite
in direction to the cur-
rent which would cause
the motion, it is evident
that the electro-magnetic ^^^- ^^- ^^^- ^^•

effect of the induced current is to oppose the motion taking
place in the field, in conformity with Lenz's law.

We have seen that the same apparatus may be used either
to transform electric into mechanical energy, or to transform
mechanical into electric energy.

521. Self-induction. — ''Extra currents.''^ — Not only does a
current at starting and stopping or changing strength act on
neighboring conductors, generating currents in them, but it
acts upon itself by a process which is called self-induction.
A current starting or increasing creates an oppositely directed
current not only in its neighbor, but also in its own wire. A
current does not start instantaneously ; it takes a certain
time — usually very short — to rise to its full strength. In
other words the circular lines of magnetic force round a
straight current do not spring into existence instantaneously,
but expand gradually like the widening ripples produced
when a stone is dropped into still water. But when started
it tends to persist, so that if its circuit be suddenly broken,
it does not stop instantaneously. The lines of force gradually

554 ETHER DYNAMICS.

collapse, but tlie point of interest is that this, collapse
gives rise to an electrical push, or E.M.F. far greater than
that which maintained the current, and this sudden drive
forward of electricity in the wire at the instant the circuit is
broken causes the spark seen on breaking a circuit ; and the
more sudden the break the more violent the spark. If a
current pass through the helix of an electro-magnet, owing to
the permeability of the iron a far larger number of lines of
force traverse its circuit than if the core were removed ; and
hence, at the stoppage of the current, a correspondingly
greater impulse operates in the wire and creates a correspond-
ingly more powerful spark. For a similar reason the self-
induction is much greater in a coil of wire than if the same
wire were laid out straight.

The self-induction at breaking a circuit is somewhat analo-
gous to the blow which a high-pressure service water-tap
experiences when a flow of water is suddenly arrested by
turning the tap. The water momentum often bursts the
pipe. Similarly self-induction often breaks through the in-
sulation of field magnets of dynamos when the current is
suddenly arrested in them.

The two effects — the delay at making circuit, and the
momentum at breaking — are frequently called '^ extra cur-
rent" effects, but they are now more commonly spoken of as
manifestations of self-induction or qiiasi-electrwal-inertia.

The action of self-induction is to hinder the sudden rise
and fall of current strength in a wire. Hence, in circuits of
large self-induction it is impossible to make very sudden
changes in the strength of a current flowing through it.
This hinders telephonic transmissions through long wires
and long coils, and renders ocean cable telegraphing a com-
paratively slow operation. It renders the changes of cur-
rents in the armatures of dynamos more sluggish than they
would otherwise be.

INDUCTION COILS.

555

522. Induction coils. — If a core of iron, or, still better, a
bundle of wires (A A, Fig. 446), be inserted in the primary
coil, it is evident that it will be magnetized and demagnetized
every time the primary is made and broken. The starting
and cessation of amperian currents in the core in the same
direction as the primary current, and simultaneously with the
commencement and ending of the primary current, greatly

Fig. 446.

intensifies the secondary current. To save the trouble of
making and breaking by hand, as in Fig. 438, the core is also
utilized in the construction of an automatic make-and-break
piece. A soft iron hammer, h, is connected with the steel
spring, c, which is in turn connected with one of the termi-
nals of the primary wire. The hammer presses against the
point of a screw, d, and thus, through the screw, closes the
circuit. But when a current passes through the primary
wire, the core becomes magnetized, draws the hammer away
from the screw, and breaks the circuit. The circuit broken,

5b6 ETHER DYNAMICS.

the core loses its magnetism, and the hammer springs back
and closes the circuit again. Thus the spring and hammer
vibrate, and open and close the primary circuit with great
rapidity. An instrument made on these principles is called
an induction coil.

523. Huhmkoi'ff^s coil. — This instrument has the impor-
tant addition to the parts already explained of a condenser,
B B. This consists of two sets of layers of tinfoil separated
by paraffined paper ; the layers are connected alternately with
one and the other electrode of the battery, as the figure shows,
so that they serve as a sort of expansion of the primary wire.
When the circuit is broken, the extra current tends to jump
across at b, and to vaporize the points of contact, and form
a bridge with the vapor of metal that would prolong the
time of breaking. But, when the condenser is attached, the
extra current finds an escape into it easier than to jump
across at b, so the vaporizing of the contact is avoided, and
the time of breaking being much shortened, the secondary is
much more intense.

The primary helices of induction coils consist of compara-
tively few turns of coarse insulated wire ; but the secondary
helices contain many turns of very fine wire, insulated with
great care. The secondary current is, at breaking, as we
ought to expect from the extreme rapidity with which the
primary circuit is broken, distinguished from the primary, or
galvanic current, by its vastly greater E.M.F., or power to
overcome resistances. A coil constructed for Mr. Spottis-
woode of London has two hundred and eighty miles of wire
in its secondary coil. With five G-rove cells this coil gives a
secondary spark forty-two inches long, and perforates glass
three inches thick. Many brilliant experiments may be per-
formed with these coils.

Experiment 8. — Connect a battery of two Bunsen cells, in multiple
arc, with a Euhmkorff coil (Fig. 447). Bring the electrodes of the sec-

DYNAMO DEFINED.

557

ondary coil within from one-fourth of an inch to one inch of each other,
according to the capacity of the
instrument. A series of sparks
in rapid succession pass from pole
to pole.

Experiment 9. — Introduce a
Geissler tube, A, into the secon-
dary circuit. These tubes con-
tain highly rarefied gases of dif-
ferent kinds. Platinum wires are
sealed into the glass at each end
to conduct the electric current
through the glass. The sparks
become diffused in these tubes so
as to illuminate the entire tubes

with an almost continuous glow. Observe that the electrodes are sepa-
rated from each other much more widely than would be admissible in
air of ordinary density, showing that rarefied gases offer less resistance
than dense gases. Gases have been so highly rarefied, however, that an
electric current would not pass.

Fig. 447.

Sectiox XIX.

DYNAMO-ELECTRIC MACHINES.

524. Dynamo defined. — "The greatest single advance in
industrial life made during the present century is the inven-
tion and perfection of the Dyyuwio- Electric Machine.

The dynamo is a device for changing mechanical energy
into electric energy. In the most improved types of dynamos
this is done with a loss of less than 5 per cent of the energy.

525. Frincijyle of the dyiiamo.'^ — The illustrations of elec-
tro-magnetic induction given in the preceding section suggest

1 Faraday's theory is this : Moving a wire across a space in which there are mag-
netic lines, so as to cut across those lines, sets up magnetic whirls around the mov-
ing wire, or, in other words, generates a so-called current of electricity in that wire.
It is necessary, however, that the moving conductor should so cut the lines of force
as to alter the number of lines of force that pass through the circuit of which the
moving conductor forms a part.

558

ETHER DYNAMICS.

the explanation of the action of a dynamo. The action of the
dynamo machine depends upon the principle that when lines
of magnetic force are cut by a wire a difference of potential
is produced in the wire, and hence a current flows through
the wire, if its ends be connected so as to form a closed cir-
cuit. This was illustrated in connection with Fig. 441, and
is illustrated by the following experiment.

Experiment 1. — Connect a flat coil of about two inches in diameter,
having several turns of wire, with a delicate galvanometer, and rotate
the coil at one of the poles of a strong magnet on an axis at right angles
to the axis of the magnet and the lines of force, as illustrated in Fig. 448.

The horizontal arrow a indicates the direction of the mag-
netic lines of force, the horizontal arrow h the direction of
motion of the end of the coil of wire, and the vertical arrow

Fig. 448.

c the direction of the current induced in the coil of wire from
the movement of the coil across the field of magnetic force in
such a manner as to cut lines of force.

If the coil be moved rapidly in front of the magnet, the
current is stronger and hence the E.M.F. must be greater
than if it be moved slowly.

Also if the number of turns of wire be increased the
E.M.F. is correspondingly increased, as will be shown by the
increased strength of current. The increase in strength of
current will not be so great as the increase of the E.M.F.

PRINCIPLE OF THE DYNAMO. 559

unless the galvanometer resistance is very large as compared
with that of the coil.

Every portion of the wire that cuts lines of force must,
then, evidently develop E.M.F., and the total E.M.F. is the
sum of all that is induced in different parts of the coil.

If the strength of the magnet be increased the number of
lines of force, and consequently the strength of current, will
be increased. Again, if the coil be of high resistance, the
current will be feeble ; if of low, the current is stronger.

We may continue our experiment still further by inserting
a bar or disk of soft iron into the coil and again moving the
end of the coil through the field of force in front of the north
pole of the magnet. A very decided increase in the strength
of current is observed. If, further, another bar magnet be
placed so that its south end faces the other end of the coil,
and the coil be fixed at its center while its two ends are
made to rotate past the two poles, more lines of force are cut
and greater E.M.F. is developed, as is seen from the increased
strength of current.

The same effect can be obtained by using a single magnet
of horse-shoe form, and placing the coil between its two
opposite poles.

We have now found that the strength of current and
E.M.F. depend upon (1) the rapidity of motion of the ivire
through the field; (2) the mimher of turns of the tvire ; and
(3) the number of lines of force cut, or the strength of field.

A powerful electro-magnet is preferable as an inducer to a
permanent magnet, as its strength or m^agnetic density can be
made much greater.

526. Relation existing between the direction of the lines of
force,' the motion of the coil, and the direction of the current. —
This relation may be conveniently remembered by placing
the thumb, forefinger, and middle finger of the right hand all
at right angles to one another, as in Fig. 449, and imagining

560

ETHER DYN^AMICS.

Fig. 449.

the forefinger to indicate the direction of the magnetic lines
of force (FOEe, FOEce), the thumb that of the motion of

^ the conducting wire

(thuMb, Motion), and
the middle finger that
of the direction of the
\L^ "-^Z induced current (mid-

dle, Induced). If the
DIRECTION of'forcb haud, wlth thc fingers
in this relative po-
sition, be held so that
the direction of the
forefinger coincides
with the direction of
the lines of force (as indicated by a test needle), and the
thumb points in the direction of motion of the part of the
conductor under consideration, the middle finger will indi-
cate the direction of the induced current.

527. The dynamo, — We are now prepared to study the
action of the dynamo. Our magnet, which is commonly an
electro-magnet, is called the
field magnet, and our coil or
series of coils of wire, which
is generally made to move in
front of the poles of the field
magnet, is called the armature.
The armature is that part of
the electric circuit in which
the induced current is gene-
rated. Like the battery, it
may be considered as the
source of the current. The number of lines of force passing
through a circuit may in general be changed in two ways :
either (1) by moving the circuit through a field in which the

Fig. 450.

THE DYNAMO.

561

density of the lines of force varies, as represented in Fig.
450 ; or (2) by rotating the plane of the circuit so as to change
the angle which it makes with the line of force, thus increasing
or decreasing the number which the circuit encloses (Fig. 443).
The former of these methods is adopted in the Thomson-Houston
and the Westinghouse alternate-current dynamos ; the latter
method is employed in the Edison and the Weston systems.
A common simple
form of dynamo is

illustrated in Fig. <^

451. A large mass
or bar of soft iron
of the U form, sur-
rounded with a coil
of insulated wire,
and terminating in
.the pole pieces N.
and S., forms the

GENERATOR

Fig. 451.

field magnet. The
armature consists

of a single rectangular loop of wire, which is fixed to a hori-
zontal axis, and terminates in two rings of metal, a and h,
which are fixed to the axle, but insulated from it.

When a current passes through the field coils, and the core
becomes magnetized, lines of force will cross and fill the space
between the pole pieces of the field magnet. As these lines
are cut by the horizontal parts of the rotating wire, an E.M.F.
is generated in these parts, and a current flows in the direc-
tion indicated by the arrows.

Apply the preceding rule for determining the direction of
the current by letting the forefinger of the right hand take
the direction of the magnetic lines of force, the thumb the
direction of motion at right angles to the lines of force, and
the middle finger the direction of the induced current. The

562

ETHER DYKAMICS.

end portions of the loop do not cut lines of force, and there-
fore no E.M.F. is generated in tliein, and they are dead wire,
— simply serving as conductors to complete the circuit. A
metallic or carbon brush, m, touches and carries off the cur-
rent from the lower horizontal segment of the rectangular
coil. This current flows through the external resistance, E,
and completes the circuit through the brush, n, to the ring, &,
and the upper half of the loop. The current will continue to
flow in this direction while the loop moves through one half
of a revolution. Since the lines of force are cut in the oppo-
site direction in the next half revolution, the current will be
reversed in the armature wire and also through the external
circuit. Thus with each half revolution of the armature a
reversal of the current takes place. This, then, would be
called an alternating curreut dynamo.

h1%. The Gomimdator. — The alternating current is not
adapted to all uses, and for many purposes it is desirable. to
have the current continuously flowing in the same direction.
To accomplish this a commutator is attached to the axis of
the armature.

In Fig. 452 the two brass rings a and h are replaced by a
single brass tube divided into two parts by cutting it length-
wise. These two segments are attached to but insulated

from the axis, and are
^ . connected with the sepa-
rate ends of the armature
wire. When the plane
of the armature coil is
perpendicular to the line
of force passing from N
to S, as in Fig. 452, no
lines of force are being
cut, and hence no E.M.F. is developed and no current flows
through the loop. But the instant it moves out of the perpen-

THE COMMUTATOR. 563

dicular in the direction of tlie arrow, lines of force will be
cut, and as the lower segment of the loop is moving upward
past the pole S, and the other segment is moving downward
in front of the pole N, a positive current flows from the loop
through the segment «, the brush m, the resistance E, the
brush n, and the strip of the commutator h. During the next
half of a revolution the lines of force will be cut from an
opposite direction by each of the horizontal segments of the
armature loop, and hence the current will be reversed. But
the segment h of the commutator will now be in contact with
the brush m; and although the current is reversed in the
armature it will flow off at the brush m as before. Inasmuch
as no E.M.r. is developed when the |)lane of the loo|) is per-
pendicular to the lines of force, it is at this point that the
brushes pass from one segment to the other. If the segments
were to leave the brushes while the coil was cutting lines of
force, it is evident that a spark would be formed if the coil
were revolving with great velocity. The maximum spark
would be given off if the segments left the brushes when the
loop is parallel to the lines of force.

Thus by means of the commutators and brushes, reversal
of the current is prevented in the external circuit, although
the current in the armature reverses with each half revolu-
tion. This arrangement would constitute a direct-current
dynamo. We may have two turns of wire before connecting
with the commutator strips, giving twice the E.M.F., or three
turns, giving three times the E.M.F. ; i.e. the E.M.F. will be
proportional to the number of turns of wire in. the coil.
Again, instead of having only one coil we may have two or
any number of coils, each separate from the others, and ter-
minating in strips or segments which are on opposite sides
of the commutator. Generally the coils are connected in
series, thus making any segment a terminal of one coil and
the beginning of the next.

564

ETHER DYNAMICS.

529. Eddy currents.

In Experiment 1 it was observed that with a mass of iron within
our coil a greater current was produced when the coil was moved
through the magnetic field. So, as in Fig. 461, the coils of wire
may be wound around a cylinder or drum of iron, winding along
the length of the drum and over the ends. From what we have
previously seen it was evident that differences of potential will be
set up in the iron, and, as it is a good conductor, currents will flow
through the iron unless prevented, and heat it. These eddy currents
(commonly called "Foucault currents") are prevented by making
the armature core laminated, i.e. having it made up of iron discs or
plates insulated from one another, i

530. The Gramme armature. — The machine we have de-
scribed has the drum or Sieiyiens armature, and is now more
commonly used. Another form of machine frequently em-
ployed has the ring or Gramme armature. In this form of

Fig. 453.

armature the core consists of an iron ring or hollow cylinder
instead of a closed cylinder or drum, and the wire is wound

1 It must be noticed that the armature current, because of its action on the core
tends to distort the magnetic field in the direction of the rotation. This distortion
necessitates a change of position or lead of the brushes on the commutator, in the
direction of the rotation of the motion of the armature.

THE GRAMME ARMATURE.

565

round this ring instead of over it. The ring may be com-
posed of a bundle of soft iron wires, as shown in Fig. 453
(where a portion of the ring is cut away), surrounded by
sectional coils of what is virtually an endless wire. A wire
from each section is carried to and connected electrically with
a section of the commutator.

Fig. 454 is a skeleton diagram of a generator of this kind ;
and Fig. 455 is a portion of the same, showing lines of force
traversing the field pieces and the armature. The lines are

Fig. 454.

supposed not to be distorted by the motion of the armature.
When the coils with the segments are connected in series
it is evident that those coils moving downward in front of
the pole piece IST (Fig. 454) develop E.M.F. urging the cur-
rent away from the brush n and towards the brush m. At
the same time those coils moving upward in front of the
pole-piece S also have currents urged towards the brush m.
Thus the armature is virtually divided into two equal parts,
each half having currents flowing to the brush m. The
brushes must therefore be placed on opposite sides of the
commutator in such a position that a line connecting them

666

ETHER DYNAMICS.

will be perpendicular to tlie lines of force. They will tlien
lead the current from the armature where the potential

Fig. 455.

difference is a maximum and from the coils that are in the
field of least action.

531. Classes of dynamos'^. — Dynamos may be divided into
different classes according to the method by which their
field magnets are excited. Eig. 456 illustrates a magneto-
electric machine, where the field magnet is a permanent steel
magnet. This form of machine is seldom used, since a per-
manent steel magnet cannot be made as powerful as an
electro-magnet having a soft iron core of equal mass.

Eig. 451 illustrates a separately excited dynamo, where the
field magnet coils receive their currents from a separate
generator, e.g. a battery, and not from the armature coils.

1 For the characteristics of tlie various classes of dynamos, as well as for a most
lucid and comprehensive treatment of dynamos generally, see Dynamo-Electric
Machinery, by S. P. Thompson.

CLASSES OF DYNAMOS.

567

Since an alternating current dynamo does not produce a con-
stant magnetic field, alternating dynamos, in general, are
separately excited. Such are the West-
inghouse and the Thomson-Houston incan-
descent-lighting dynamos.

Fig. 454 is a series dynamo, where the
coils of the field magnet are joined in
series with the armature so that the en-
tire current passes through these coils.

Fig. 457 illustrates a shu7it machine,
where the field-coil serves as a shunt to
the external circuit. L is the main wire
and I is the shunt wire. In the shunt

machine
only a part

of the current generated in
the armature passes through
the field-coils.

A dynamo is said to be
" self-exciting " when the
whole (Fig. 454) or any part
(Fig. 457) of the current which
is produced is used to magnet-
ize the field magnets. Such
are the Edison incandescent
dynamos.

The fields, after being
once excited from any
source, e.g. another dynamo,
always retain a little re-
sidual magnetism, so that
when the armature begins
to rotate, a slight current
is at once induced in it.

568

ETHER DYNAMICS.

This strengthens the field, and the stronger field reacts to
increase the current, so that the current soon rises to its
normal strength.

More than one set of coils may be used on the field mag-
nets, and these coils may receive currents from different
sources. Dynamos employing these are called compound
wound machines. They may be arranged for constant poten-
tial, or constant current. A combination of the series and
separately excited, or of series and shunt, gives constant
potential ; while a combination of shunt and separately ex-
cited gives constant current.

All figures hitherto have been diagrammatic representations
of dynamos. Fig. 458 represents a modern typical dynamo,

Fig. 458.

the Weston. Large field magnets, A and B, are placed each
side of the revolving armature. A steam-engine communi-
cates motion to the armature by means of a belt passing over
the circumference of the wheel W. The pole-pieces, as will

CLASSES OF DYNAMOS.

569

be seen in the cut, are laminated to prevent eddy-currents,
and the magnets are shunt-wound.

Fig. 459 represents one of the most common forms of the
Edison dynamo, and Fig. 460 is a skeleton diagram correspond-

FlG. 459.

ing in most particulars with the first. It will be seen by the
latter figure that it is a shunt-wound dynamo. The terminals
of an automatic regulator for regulating the intensity of the
current are inserted in the binding screws a a. P is a so-called
pilot-lamp joined in multiple arc to the field-coils. FF are
leading wires ; and h h are points for the attachment of fuses.
These fuses are to the dynamo what the safety-valve is to the

570

ETHER DYNAMICS.

steam boiler ; they protect the dynamo from injury by over-
pressure, since an overload is sure to cause them to melt and
thus interrupt the current.

532. Classes of armatures} — (1) In ring-armatures the
coils are wound round a ring-shaped core, as shown in Fig.
454. Example : the Gramme and the Brush.

(2) In drum-armatures the coils are wound longitudinally

Fig. 460.

Fig. 461.

over a cylinder or drum, as in Fig. 461. Examples : the
Edison, the Weston, and the Siemens.

(3) In pole or radial armatures the coils are wound on sep-
arate poles that project radially from a cylinder (Fig. 462).

In alternating current dynamos, in order to obtain the rapid
reversals (in some machines as many as 200 per second) of
currents in opposition to resistance offered by self-induction,
a number of poles of alternate polarity are employed.

The separate coils may be coupled either in series or in
multiple-arc. When low E.M.F. is desired, as for incan-
descent lamps in multiple arc, the separate coils are united

1 The Thomson-Houston armature cannot be classified, as it is unique among
armatures. It is spheroidal in shape.

REVERSIBILITY OF THE DYNAMO.

571

in multiple arc ; but where great E.M.F. is required, they
are connected in series, as shown in Figs. 462 and 463.

Fig. 463.

(4) Disk-armatures are usually composed of a number of
separate coils set side by side in the circumference of a disk
(Fig. 463). Mechanical difficulties in their construction have
not permitted them as yet to compete successfully with the
first two types named above.

Section XX.

ELECTRIC MOTOR.

533. Reversibility of the dynamo. — This subject has al-
ready been touched upon in §§ 512 and 520 ; it only remains
to treat it a little more in detail. If a current from an
external source, e.g. a battery or another dynamo, be passed
through the armature and field magnet of a direct-current
dynamo, it will excite the armature and make of it an electro-
magnet and will also excite the fields. The current will enter
at the terminals and will pass through the commutator into
the Armature. The relation of parts is such that in doing
this it will develop N and S poles in parts of the periphery
of the armature distant from the N and S poles of the fields.
Hence a stress will be set up between the armature and the
poles of the field magnet tending to move the former a little

572 ETHER DYNAMICS.

in the opposite direction to that in which it is compelled to
move when generating a current. But as soon as it has
turned a short distance, the action of the commutator shifts
the current, and new poles are established in the armature
back of the first and in the same relative positions which they
at first occupied. The armature continues to rotate as the
new poles are attracted and repelled, and the action goes on
so long as a current is supplied. Obviously if there were no
commutator the poles of the armature would be fixed, and it
never could rotate through a greater angle than 180°.

It is evident, then, that if two dynamos be connected by
wires in the same circuit and if the armature of one be
rotated, the armature of the other will rotate in a reverse
direction as soon as the current transmitted from the first
attains a certain intensity.

If in a separately-excited direct-current dynamo, or in a
magneto-dynamo (Fig. 456), or in a series dynamo (Fig. 454),
we substitute a dynamo or other current generator for the
resistance E, and the current be made to flow through the.
armature in the same direction as Avhen generating a current
as shown by the arrows, the armature will tend to rotate in
the opposite direction from that indicated by the barbed ar-
rows. If the current from our generator flows in the opposite
direction from that shown by the arrows, the armature
will rotate in the same direction as the barbed arrow. If
again we use a shunt dynamo (Fig. 457), placing a generator
in the main circuit L with a current flowing into the armature
in the direction of the arrow, the current will divide at the
negative brush, a part going through the field-magnet coils,
but in a direction opposite to that in which it flows wh^ the
dynamo is used as a generator. Hence the polarity of the
field-magnet is reversed, and the armature will rotate in the
same direction when receiving a current as when generating
a similar current.

THE ACTION OF THE DYNAMO-MOTOR.

573

It is now evident that the dynamo is a reversible machine,
in which mechanical energy can be changed directly into
electrical energy or electrical energy into mechanical energy.
When the dynamo is used for the latter transformation, it is
commonly known as an elective motor. In other words a
modern motor is a dynamo reversed. The discovery of the
reversibility of the dynamo is considered to be one of high
importance. The reversibility leads to some curious results.
For example, when a car on an electric railway is descending
a hill, its motor, instead of driving the car, might be driven
by it and thereby become a dynamo and send a current into
the line to drive or help drive other cars in the same circuit.
It would be possible for one car of an electric system, in
running down a steep hill, to have its mechanical energy
absorbed by its motor acting as a dynamo (and thus serving
as a brake to retard its motion), and thus to draw another car
of the same system up a hill miles distant.

534. The actloji of the dynamo-motor. — This may be under-
stood by referring to Fig. 464, and imagining a generator
to replace the external resistance E. Suppose the current
from the generator enters

at the brushes and flows . ,1^^

in the loop in the direc-
tion of the arrows ; then
the upper face of the loop
will have S polarity and
the under face N polarity.
Then by the mutual action
between this field and
that of the magnet N S, a rotation of the loop will take place
clockwise till it comes into a vertical position. When it
reaches this position, however, the brushes are so arranged
with reference to the commutator segments that the current
in the loop — and hence its polarity — is reversed. Even if

574

ETHER DYNAMICS.

there were only one loop its inertia would be sufficient to
carry it by this critical position, and the loop would continue
to rotate in the attempt again to bring its field parallel to that
of N S ; but as a matter of fact the other loops in the
armature are never in the critical position at the same time
as the one considered, and those on each side of it conspire
to produce a continuous rotation in the same direction.

If the armature contain a soft iron core, as is usually the
case, the intensity of the field will be much greater and the
mechanical effect correspondingly increased.

Fig. 465 represents a modern form of motor weighing only
two or three pounds, and capable, when worked with four or

five Bunsen cells, of
operating a sewing
machine or running a
small saw. It consists
of a movable coil with-
in a fixed coil. The
wires of each coil are
wound on an iron
frame-work, the two
opposite edges of the
iron being north and south poles when the current is passing.
The inner coil is furnished with a commutator, which
reverses the current as
soon as opposite poles
of the inner and outer
coils are opposed. A
represents the outer
coil of wires, B one
pole of the fixed electro-
magnet made by it, and C the commutator by which the inner
coil has the current reversed each half revolution. Fig. 466
shows the inner coil (D), whose terminals are attached to the

Fig. 465.

Fig. 466.

THE INDUCTION COIL REVERSIBLE. 575

two halves of the spindle (E), which are carefully insulated
from each other. In Fiof. 467 the commutator is shown in
plan. The current enters the inner coil through the spring H,
which carries a friction roller ^^_

working on the commutator E ;
after traversing the coil it re-
turns to the upper half of E,
and thence passes by the spring
Gr to K, from K through the
outer coil to L, and from L
back to the battery. fig. 467.

The dynamo as a generator and the dynamo as a motor
have already revolutionized electrical economics and relegated
the battery to an honored position among things of the past.
The electric motor is now extensively used in large towns
and cities, in factories where power is not continuously
needed. Its widest application at present is in the propul-
sion of street cars. The current is generated by dynamos at
some central power house and thence distributed to the
motors at various points on the circuit.

Section XXI.

_ THE TRANSFORMER.

535. The induction coil reversible. — An induction coil is
in a certain sense a reversible machine. If a current of
considerable strength circulate under small E.M.F. in the
primary, then variations in its strength give rise to very
weak currents of exceedingly high E.M.F. in the secondary.
Conversely, if we cause to circulate in the secondary weak
currents under very high E.M.F., by their fluctuations there
will be generated in the primary strong currents of small
E.M.F. We do not in either case create electric energy.

576 ETHER DYNAMICS.

Electric activity (or power) is the product of two factors,
current and electro-motive force. The induction coil enables
us to increase one of these factors at the expense of the
other, and to transform electric energy in form much as a
mechanical power (e.g. a lever) enables us to convert a quan-
tity of work which consists of small stress exerted through
a great distance into a large stress exerted through a small
distance.

The transformer — sometimes called a converter — is merely
an induction coil used to change the relation of the number
of volts to the number of amperes of any current. In a
perfect transformer the number of watts in the primary
equals the number of watts in the secondary.

The Euhmkorff coil as ordinarily used may be regarded as
a ^^ step up " transformer from low potential to high poten-
tial. But if the coil of long thin wire be used as the primary,
it becomes a "step down" transformer from high potential
to low potential.

If we have a primary consisting of two thousand turns of
wire and a secondary of one thousand turns, we get in our
secondary half the voltage of the primary, but twice the
number of amperes. In general, if we could concentrate all
the lines of force of the primary upon the secondary we
should have the following relation : volts • in primary X
amperes in primary = volts in secondary X amperes in sec-
ondary.

Fig. 468 represents the coils of a transformer used in the
incandescent lamp service, and Fig. 469 is the same enclosed
in a case. These transformers are usually supported on the
street poles.

The transformer is applied in the welding of metals, i.e. to
fuse the ends of metals that are to be joined together, where
many hundred or even thousand amperes of current, and only
a fraction of a volt, would be required for an instant.

THE INDUCTION COIL REVERSIBLE.

577

A still wider application of transformers is in tlie trans-
mission of electric activity.

Since the rate at which a current performs work equals
the volts times the amperes (E X C), then according to Ohm's
law the work done per second by a current passing through
a conductor equals C^R (§ 463). That is, when the current

Fig. 468.

Fig. 469.

strength is doubled there will be four times as much energy
transformed per second. We see, then, that to transfer electric
energy to a great distance it may be desirable to have a high
E.M.F. with a small current passing through the mains, and
then to reduce the E.M.F. and increase the current by a
transformer at the place where the energy is to be used. By
this means the expense involved in the copper conductors is
much reduced.

For electric lighting in private houses transformers are
used to bring down the high potential of the mains to the
safe limit of about 100 volts.

The transformers may be arranged in series, as shown in
Fig. 470, in which case the same primary current passes

678

ETHER DYNAMICS.

through all of them, and from each secondary circuit leads
may be taken off to work local lights. Or they may be

Fig. 470.

arranged in multiple arc (Fig. 471), in which case each
primary current is a bridge across from mam to main of the

MAIN CONDUCTOR

Fig. 471.

REVERSIBILITY OF ELECTROLYSIS. 579

dynamo which supplies the alternating current of high
E.M.F. In both cases the secondary current would, if used
for lighting incandescent lamps, be led through lamps in
multiple arc.

Section XXII.

SECONDARY OR STORAGE BATTERIES.

536. Reversihility of electrolysis. — If water be decomposed
for a time between neutral electrodes such as platinum plates
and then the battery or other generator be withdrawn from
the circuit and replaced by a sensitive galvanometer, a
deflection of the needle shows that a transitory current flows
in the opposite direction to the primary or electrolyzing cur-
rent. It is evident that the electrolyzing current polarizes
the electrodes in the electrolyte, and that energy is thus
stored in the cell. When the wires are joined, this polariza-
tion causes a current to flow during an appreciable period,
and the platinum electrodes become depolarized. The elec-
trical energy of the cell is converted into chemical potential
energy in that it overcomes the E.M.F. of the decomposing
cell. Polarization is of the nature of a counter E.M.F. It
is precisely this polarization which we have to contend with
in nearly all voltaic cells (§ 437), and which we seek to neu-
tralize by means of depolarizing substances.

Devices for thus storing up energy by electrolysis, and
liberating it when desired in the form of electric current, are
called storage or secondary batteries, and sometimes accumu-
lators. Note that the process is an electrical storage of
energy, not a storage of electricity. The energy assumes
the form of chemical potential energy, and there is really
no more electricity in the cell when it is fully charged than
at the commencement of the operation.

580 ETHER DYNAMICS.

If, instead of platinum electrodes, two plates of lead cov-
ered with a coating of red lead, or, better, the positive plate
(which is the positive electrode when the cell is being
charged) covered with a paste of red lead and sulphuric
acid, and the negative plate with a paste of litharge and
sulphuric acid, be used as electrodes, dipping as before into
sulphuric acid, and the electrodes be connected with a pow-
erful voltaic battery (or, better, with a dynamo), the positive
electrode becomes by electrolysis peroxydized by the oxygen
which is liberated, while the negative is deoxydized by the
hydrogen. The plates may remain in this state for many
days. Hence the storage battery is a very convenient means
of accumulating energy at one time or place, and using it at
some other time or place. For example, energy may be stored
during the daytime ; and this energy, reconverted into electric
energy, may feed incandescent lamps at night at any conven-
ient place. Or these batteries, having been charged by a
dynamo, may be transported to lecture-halls, workshops, elec-
tric cars, etc., where powerful currents may be needed. The
E.M.F. of these batteries may be multiplied many-fold by
joining them in series on the same principle as the E.M.F.
of voltaic batteries is increased. The E.M.F. of a single cell
similar to the above is about 2.2 volts. The internal resist-
ance of a cell whose surface of electrodes is 300 cm^ is about
.006 ohm. Their low resistance constitutes one of their chief
virtues as generators. An idea of the capability of a storage
battery may be obtained from the statement that a battery
capable of furnishing one horse-power for five hours weighs
500 lbs. ; it will supply twelve incandescent lamps of sixteen
candle-power each for five hours. It will then require to be
recharged. The great fault of these accumulators in their
present form is their want of durability.

ELECTRICAL TRANSMISSION OF ACTIVITY. 581

Section XXIII.

ELECTRICAL TRANSMISSION OF ACTIVITY.

537. Essentials for the tixinsmission of activity electrically. —
The electrical transmission of activity (or power) demands
(1) a source of energy (e.g. a water-power or steam engine)
and a dynamo ; (2) a line of wire to serve as a conductor of
electricity ; and (3) a motor. The current generated in the
dynamo reaches the motor (which may be many miles away)
and causes a rotation, and the activity is communicated there-
from to machinery. Sometimes the circuit is metallic through-
out, but often, as in the case of the electric railroad, the earth
is used in place of a return wire.

538. Limitations. — Activity of any desired magnitude may
be transmitted, provided the size of the conductor be not too
small for the current employed. The rate at which energy
is expended in the motor is proportional to the potential
difference at its terminals, and to the current strength. If
the dynamo give a high potential difference, less current will
be required to furnish a required activity. The smaller
current heats the wire less (Why? See Section VIL); con-
sequently the wire may be smaller as the motor works at a
higher difference of potential. The size of the wire that
should be used depends wholly upon the strength of the
current to be transmitted. For example, the entire energy of
Niagara Falls might be transmitted by an ordinary telegraph
wire were it not that the enormous potential difference re-
quired would cause a leakage of current at every possible
avenue of escape, and therefore make the system a menace
to every one near it.

539. Electric railiuays. — These furnish the most familiar
illustrations of transmission of electric activity. The current
from a dynamo stationed at some " electric plant " is conveyed

582 ETHER DYNAMICS.

by a trolley wire running along the road. Each car carries
one or more motors, one of whose terminals connects with
the trolley and the other with the earth through the wheels
and rails. One ot^the terminals of the dynamo is also carried
to the earth. Cars on an electric railway are usually worked
in multiple arc. The trolley wire and the earth being re-
garded as parallel conductors, the car motors serve as bridges
from one to the other, like the rungs of a ladder.

540. Advantages of electrical transmissio7i of activity. The
advantages of this method of transmission of activity over
that by means of belts, shafts, compressed air, etc., are many
and important. For example, it is attended usually with
much less waste of energy. An electrical conductor is flex-
ible ; it can be bent and carried round corners and tapped
wherever wanted. It is motionless, though transmitting
large quantities of energy. It transmits energy through very
long distances.^

Section XXIV.

THERMO-ELECTEIG CURRENTS.

541. Heat energy transformed directly into electric energy.

Experiment 1. — Insert in one screw-cup of a sensitive galvanometer
an iron wire, and in the otlier cup a copper, or better, a German-silver
wire. Twist the other ends of the wire together, and heat them at their
junction in a flame ; a deflection of the needle shows that a current of
electricity is traversing the wire. Place a piece of ice at their junction ;
a deflection in the opposite direction shows that a current now traverses
the wire in the opposite direction.

Experiment 2. — Take a strip of sheet copper about 15 inches long
and three-fourths of an inch wide, and a strip of zinc of the same dimen-

1 In some places, in the mining regions of the Western States this is the only
practicable way of supplying activity to the crushers and stamps of the mills. These
could no